Expressing Combinatorial Optimization Problems by
Systems of Polynomial Equations and the
Nullstellensatz
J . A . D E L O E R A1†, J . L E E2 , S . M A R G U L I E S3† and S . O N N4‡
1
2
3
Department of Mathematics, Univ. of California, Davis, California, USA
(email: deloera@math.ucdavis.edu)
IBM T.J. Watson Research Center, Yorktown Heights, New York, USA
(email: jonlee@us.ibm.com)
Department of Computer Science, Univ. of California, Davis, California, USA
(email: smargulies@ucdavis.edu)
4 Davidson
Faculty of IE & M, Technion - Israel Institute of Technology, Haifa, Israel
(email: onn@ie.technion.ac.il)
Systems of polynomial equations over the complex or real numbers can be used to model combinatorial
problems. In this way, a combinatorial problem is feasible (e.g. a graph is 3-colorable, hamiltonian, etc.)
if and only if a related system of polynomial equations has a solution. In the first part of this paper, we
construct new polynomial encodings for the problems of finding in a graph its longest cycle, the largest
planar subgraph, the edge-chromatic number, or the largest k-colorable subgraph.
For an infeasible polynomial system, the (complex) Hilbert Nullstellensatz gives a certificate that the
associated combinatorial problem is infeasible. Thus, unless P = NP , there must exist an infinite sequence
of infeasible instances of each hard combinatorial problem for which the minimum degree of a Hilbert
Nullstellensatz certificate of the associated polynomial system grows.
We show that the minimum-degree of a Nullstellensatz certificate for the non-existence of a stable set of
size greater than the stability number of the graph is the stability number of the graph. Moreover, such a
certificate contains at least one term per stable set of G . In contrast, for non-3-colorability, we found only
graphs with Nullstellensatz certificates of degree four.
1. Introduction
N. Alon [1] used the term “polynomial method” to refer to the use of non-linear polynomials for solving
combinatorial problems. Although the polynomial method is not yet as widely used by combinatorists as, for
instance, polyhedral or probabilistic techniques, the literature in this subject continues to grow. Prior work on
encoding combinatorial properties included colorings [2, 9, 10, 15, 24, 27, 28, 29], stable sets [9, 23, 24, 36],
matchings [11], and flows [2, 29, 30]. Non-linear encodings of combinatorial problems are often compact.
This contrasts with the exponential sizes of systems of linear inequalities that describe the convex hull
of incidence vectors of many combinatorial structures (see [37]). In this article we present new encodings
for other combinatorial problems, and we discuss applications of polynomial encodings to combinatorial
optimization and to computational complexity.
†
‡
Research supported in part by an IBM Open Collaborative Research Award and by NSF grant DMS-0608785
Research supported by the ISF (Israel Science Foundation) and by the fund for the promotion of research at Technion
2
J.A. De Loera, J. Lee, S. Margulies, S. Onn
Recent work demonstrates that one can derive good semidefinite programming relaxations for combinatorial optimization problems from the encodings of these problems as polynomial systems (see [22] and
references therein for details). Lasserre [20], Laurent [21] and Parrilo [31, 32] studied the problem of minimizing a general polynomial function f (x) over an algebraic variety having only finitely many solutions.
Laurent proved that when the variety consists of the solutions of a zero-dimensional radical ideal I , there
is a way to set up the optimization problem min{f (x) : x ∈ variety(I)} as a finite sequence of semidefinite
programs terminating with the optimal solution (see [21]).
This immediately suggests an application of the polynomial method to combinatorial optimization problems: Encode your problem with polynomials equations in R[x1 , . . . , xn ] that generate a zero-dimensional
(variety is finite) radical ideal, then generate the finite sequence of SDPs following the method in [21]. This
highlights the importance of finding systems of polynomials for various combinatorial optimization problems. The first half of this paper proposes new polynomial system encodings for the problems, with respect
to an input graph, of finding a longest cycle, a largest planar subgraph, a largest k-colorable subgraph, or a
minimum edge coloring. In particular, we establish the following result.
Theorem 1.1.
1. A simple graph G with nodes 1, . . . , n has a cycle of length L if and only if the following zero-dimensional
system of polynomial equations has a solution:
n
X
yi = L .
(1.1)
i=1
For every node i = 1, . . . , n:
n
Y
yi (yi − 1) = 0,
(xi − s) = 0 ,
(1.2)
s=1
yi
Y
(xi − yj xj + yj )(xi − yj xj − yj (L − 1)) = 0 .
(1.3)
j∈Adj(i)
Here Adj(i) denotes the set of nodes adjacent to node i .
2. Let G be a simple graph with n nodes and m edges. G has a planar subgraph with K edges if and only if
the following zero-dimensional system of equations has a solution:
For every edge {i, j} ∈ E(G):
X
2
z{ij}
− z{ij} = 0,
z{ij} − K = 0 .
(1.4)
{i,j}∈E(G)
For k = 1, 2, 3 , every node i ∈ V (G) and every edge {i, j} ∈ E(G):
n+m
Y
(x{i}k − s) = 0,
s=1
n+m
Y
(y{ij}k − s) = 0 ,
(1.5)
s=1


 Y ¡
¢
sk 
x{i}k − x{j}k

i,j∈V (G)
i<j
Y
i∈V (G),
{u,v}∈E(G)
¡
x{i}k − y{uv}k
¢
Y
¡
¢
y{ij}k − y{uv}k 
=1.
(1.6)
{i,j},{u,v}∈E(G)
For k = 1, 2, 3 , and for every pair of a node i ∈ V (G) and incident edge {i, j} ∈ E(G):
¡
¢
z{ij} y{ij}k − x{i}k − ∆{ij,i}k = 0 .
(1.7)
Expressing Combinatorial Problems by Polynomial Equations
For every pair of a node i ∈ V (G) and edge {u, v} ∈ E(G) that is not incident on i:
¡
¢¡
¢¡
¢
z{uv} y{uv}1 − x{i}1 − ∆{uv,i}1 y{uv}2 − x{i}2 − ∆{uv,i}2 y{uv}3 − x{i}3 − ∆{uv,i}3 = 0 ,
¡
¢¡
¢¡
¢
z{uv} x{i}1 − y{uv}1 − ∆{i,uv}1 x{i}2 − y{uv}2 − ∆{i,uv}2 x{i}3 − y{uv}3 − ∆{i,uv}3 = 0 .
3
(1.8)
(1.9)
For every pair of edges {i, j}, {u, v} ∈ E(G) (regardless of whether or not they share an endpoint):
¡
¢¡
¢¡
¢
z{ij} z{uv} y{ij}1 −y{uv}1 −∆{ij,uv}1 y{ij}2 −y{uv}2 −∆{ij,uv}2 y{ij}3 −y{uv}3 −∆{ij,uv}3 = 0 , (1.10)
¡
¢¡
¢¡
¢
z{ij} z{uv} y{uv}1 −y{ij}1 −∆{uv,ij}1 y{uv}2 −y{ij}2 −∆{uv,ij}2 y{uv}3 −y{ij}3 −∆{uv,ij}3 = 0 . (1.11)
For every pair of nodes i, j ∈ V (G) , (regardless of whether or not they are adjacent):
¡
¢¡
¢¡
¢
x{i}1 − x{j}1 − ∆{i,j}1 x{i}2 − x{j}2 − ∆{i,j}2 x{i}3 − x{j}3 − ∆{i,j}3 = 0 ,
¡
¢¡
¢¡
¢
x{j}1 − x{i}1 − ∆{j,i}1 x{j}2 − x{i}2 − ∆{j,i}2 x{j}3 − x{i}3 − ∆{j,i}3 = 0 .
(1.12)
(1.13)
For every ∆index (e.g., ∆{ij,uv}k , ∆{ij,i}k , etc.) variable appearing in the above system:
n+m−1
Y
¡
¢
∆index − d = 0 .
(1.14)
d=1
3. A graph G has a k-colorable subgraph with R edges if and only if the following zero-dimensional system
of equations has a solution:
X
yij − R = 0 .
(1.15)
{i,j}∈E(G)
For every vertex i ∈ V (G):
xki = 1 .
(1.16)
For every edge {i, j} ∈ E(G):
2
yij
− yij = 0,
¡
¢
yij xk−1
+ xk−2
xj + · · · + xk−1
=0.
i
i
j
(1.17)
4. Let G be a simple graph with maximum vertex degree ∆ . The graph G has edge-chromatic number ∆ if
and only if the following zero-dimensional system of polynomials has a solution:
For every edge {i, j} ∈ E(G):
x∆
ij = 1 .
For every node i ∈ V (G):



si 

(1.18)
Y

(xij − xik )
=1,
(1.19)
j,k∈Adj(i)
j<k
where Adj(i) is the set of nodes adjacent to node i .
[By Vizing’s theorem, if the system has no solution, then G has edge-chromatic number ∆ + 1 .]
In the second half of the article, we look at the connection between polynomial systems and computational
complexity. We have already mentioned that semidefinite programming is one way to approach optimization.
It is natural to ask how big are such SDPs. For simplicity of analysis, we look at the case of feasibility instead
of optimization. In this case, the SDPs are replaced by a large-scale linear algebra problem. We will discuss
details in Section 3. For a hard optimization problem, say Max-Cut, we associate a system of polynomial
equations J such that the system has a solution if and only if the problem has a feasible solution. On
the other hand, the famous Hilbert Nullstellensatz (see [7]) states that a system of polynomial equations
4
J.A. De Loera, J. Lee, S. Margulies, S. Onn
J = {f1 (x) = 0, f2 (x) = 0, . . . , fr (x) = 0} with complex coefficients has no solution in Cn if and only if there
P
exist polynomials α1 , . . . , αr ∈ C[x1 , . . . , xn ] such that 1 =
αi fi . Thus, if the polynomial system J has
no solution, there exists a certificate that the combinatorial optimization problem is infeasible.
There are well-known upper bounds for the degrees of the coefficients αi in the Hilbert Nullstellensatz
certificate for general systems of polynomials, and they turn out to be sharp (see [18]). For instance, the
following well-known example shows that the degree of α1 is at least dm :
f1 = xd1 , f2 = x1 − xd2 , . . . , fm−1 = xm−2 − xdm−1 , fm = 1 − xm−1 xd−1
.
m
But polynomial systems for combinatorial optimization are special. One question is how complicated are
the degrees of Nullstellensatz certificates of infeasibility? As we will see in Section 3, unless P = NP , for
every hard combinatorial problem, there must exist an infinite sequence of infeasible instances for which the
minimum degree of a Nullstellensatz certificate, for the associated system of polynomials, grows arbitrarily
large. This was first observed by L. Lovász who proposed the problem of finding explicit graphs in [24]. A
main contribution of this article is to exhibit such growth of degree explicitly. In the second part of the paper
we discuss the growth of degree for the NP-complete problems stable set and 3-colorability. We establish the
following theorem:
Theorem 1.2.
1. Given a graph G , let α(G) denote its stability number. A minimum-degree Nullstellensatz certificate for
the non-existence of a stable set of size greater than α(G) has degree equal to α(G) and contains at least
one term per stable set in G .
2. Every Nullstellensatz certificate for non-3-colorability of a graph has degree at least four. Moreover, in
the case of a graph containing an odd-wheel or a clique as a subgraph, a minimum-degree Nullstellensatz
certificate for non-3-colorability has degree exactly four.
The paper is organized as follows. Our encoding results for longest cycle and largest planar subgraph
appear in Subsection 2.1. As a direct consequence, we recover a polynomial system characterization of the
hamiltonian cycle problem. Similarly, we discuss how to express, in terms of polynomials, the decision question
of whether a poset has dimension p . The encodings for edge-chromatic number and largest k-colorable
subgraph also appear in Subsection 2.1. As we mentioned earlier, colorability problems were among the first
studied using the polynomial method; we revisit those earlier results and end Subsection 2.2 by proposing a
notion of dual coloring derived from our algebraic set up. In Section 3 we discuss how the growth of degree
in the Nullstellensatz occurs under the assumption P 6= NP . We also sketch a linear algebra procedure
we used to compute minimum-degree Nullstellensatz certificates for particular graphs. In Subsection 3.1
we demonstrate the degree growth of Nullstellensatz certificates for the stable set problem. In contrast, in
Subsection 3.2, we exhibit many non-3-colorable graphs where there is no growth of degree.
2. Encodings
In this section, we focus on how to find new polynomial encodings of some combinatorial optimization
problems. We begin by recalling two nice results in the polynomial method that will be used later on. D.
Bayer established a characterization of 3-colorability via a system of polynomial equations [4]. We generalize
Bayer’s result as follows:
Lemma 2.1.
The graph G is k-colorable if and only if the following zero-dimensional system of equations
xk−1
i
+
xk−2
xj
i
xki − 1 = 0,
+ · · · + xk−1
= 0,
j
for every node i ∈ V (G),
for every edge {i, j} ∈ E(G) ,
Expressing Combinatorial Problems by Polynomial Equations
5
has a solution. Moreover, the number of solutions equals the number of distinct k-colorings multiplied by k! .
Recall that a stable set or independent set in a graph G is a subset of vertices such that no two vertices
in the subset are adjacent. The maximum size α(G) of a stable set is called the stability number of G . We
view the stable sets in terms of their incidence vectors. These are 0/1 vectors of length |V | , one for every
stable set, where a one in the i-th entry indicates that the i-th vertex is a member of the associated stable
set. These 0/1 vectors can be fully described by a small system of quadratic equations:
Lemma 2.2 (L. Lovász [24]). The graph G has stability number at least k if and only if the following
zero-dimensional system of equations
x2i − xi = 0,
xi xj
n
X
= 0,
for every node i ∈ V (G),
(2.1)
for every edge {i, j} ∈ E(G),
(2.2)
xi = k,
(2.3)
i=1
has a solution.
Example 2.3.
Consider the Petersen graph labeled as in Figure 1. If we wish to check whether there are
1
6
2
3
5
8
9
10
7
4
Figure 1. Petersen graph
stable sets of size four, we take the ideal I generated by the polynomials in Eq. 2.1, 2.2 and 2.3:
­
I = x21 − x1 , x22 − x2 , x23 − x3 , x24 − x4 , x25 − x5 , x26 − x6 , x27 − x7 , x28 − x8 , x29 − x9 , x210 − x10 ,
x1 x6 , x2 x8 , x3 x10 , x4 x7 , x5 x9 , x1 x2 , x2 x3 , x3 x4 , x4 x5 , x1 x5 , x6 x7 , x7 x8 , x8 x9 , x9 x10 , x6 x10 ,
®
x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 − 4 .
By construction, we know that the quotient ring R := C[x1 , . . . , x10 ]/I is a finite-dimensional C-vector
space. Because the ideal I is radical, its dimension equals the number of stable sets of cardinality four in
the Petersen graph (not taking symmetries into account). Using Gröbner bases, we find that the monomials
1, x10 , x9 , x8 , x7 form a vector-space basis of R and that there are no solutions with cardinality five; thus
α(Petersen) = 4 . It is important to stress that we can recover the five different maximum-cardinality stable
sets from the knowledge of the complex finite-dimensional vector space basis of R (see [8]).
Next we establish similar encodings for the combinatorial problems stated in Theorem 1.1.
2.1. Proof of Theorem 1.1
Proof (Theorem 1.1, Part 1). Suppose that a cycle C of length L exists in the graph G . We set yi = 1
or 0 depending on whether node i is on C or not. Next, starting the numbering at any node of C , we set
xi = j if node i is the j-th node of C . It is easy to check that Eqs. 1.1 and 1.2 are satisfied.
To verify Eq. 1.3, note that since C has length L , if vertex i is the j-th node of the cycle, then one of its
neighbors, say k , must be the “follower”, namely the (j + 1)-th element of the cycle. If j < L , then the
6
J.A. De Loera, J. Lee, S. Margulies, S. Onn
factor (xi − xk − 1) = 0 appears in the product equation associated with the i-th vertex, and the product is
zero. If j = L , then the factor (xi − xk − (L − 1)) = 0 appears, and the product is again 0. Since this is true
for all vertices that are turned “on”, and for all vertices that are “off”, we have Eq. 1.3 automatically equal
to zero, all of the equations of the polynomials vanish.
Conversely, from a solution of the system above, we see that L variables yi are not zero; call this set C .
We claim that the nodes i ∈ C must form a cycle. Since yi 6= 0 , the polynomial of Eq. 1.3 must vanish; thus
for some j ∈ C ,
(xi − xj + 1) = 0 ,
or (xi − xj − (L − 1)) = 0 .
Note that Eq. 1.3 reduces to this form when yi = 1 . Therefore, either vertex i is adjacent to a vertex j (with
yj = 1) such that xj equals the next integer value (xi + 1 = xj ), or xi − L = xj − 1 (again, with yj = 1).
In the second case, since xi and xj are integers between 1 and L , this forces xi = L and xj = 1 . By the
pigeonhole principle, this implies that all integer values from 1 to L must be assigned to some node in C
starting at vertex 1 and ending at L (which is adjacent to the node receiving 1).
We have the following corollary.
Corollary 2.4. A graph G has a hamiltonian cycle if and only if the following zero-dimensional system of
2n equations has a solution. For every node i ∈ V (G) , we have two equations:
n
Y
s=1
(xi − s) = 0 ,
and
Y
(xi − xj + 1)(xi − xj − (n − 1)) = 0 .
j∈Adj(i)
The number of hamiltonian cycles in the graph G equals the number of solutions of the system divided by
2n .
Proof. Clearly when L = n we can just fix all yi to 1 , thus many of the equations simplify or become
obsolete. We only have to check the last statement on the number of hamiltonian cycles. For that, we remark
that no solution appears with multiplicity because the ideal is radical. That the ideal is radical is implied by
the fact that every variable appears as the only variable in a unique square-free polynomial (see page 246 of
[19]). Finally, note for every cycle there are n ways to choose the initial node to be labeled as 1 , and then
two possible directions to continue the labeling.
Note that similar results can be established for the directed graph version, thus one can consider paths
or cycles with orientation. Also note that, we can use the polynomials systems above to investigate the
distribution of cycle lengths in a graph (similarly for path lengths and cut sizes). This topic has several
outstanding questions. For example, a still unresolved question of Erdös and Gyárfás [35] asks: If G is a
graph with minimum-degree three, is it true that G always has a cycle having length that is a power of two?
Define the cycle-length polynomial as the square-free univariate polynomial whose roots are the possible
cycle lengths of a graph (same can be done for cuts). Considering L as a variable, the reduced lexicographic
Gröbner basis (with L the last variable) computation provides us with a unique univariate polynomial on L
that is divisible by the cycle-length polynomial of G .
Now we proceed to the proof of part 2 of Theorem 1.1. For this we recall Schnyder’s characterization of
planarity in terms of the dimension of a poset [33]: For an n-element poset P , a linear extension is an order
preserving bijection σ : P → {1, 2, . . . , n} . The poset dimension of P is the smallest integer t for which there
exists a family of t linear extensions σ1 , . . . , σt of P such that x < y in P if and only if σi (x) < σi (y) for all
σi . The incidence poset P (G) of a graph G with node set V and edge set E is the partially ordered set of
height two on the union of nodes and edges, where we say x < y if x is a node and y is an edge, and y is
incident to x .
Expressing Combinatorial Problems by Polynomial Equations
7
Lemma 2.5 (Schnyder’s theorem [33]). A graph G is planar if and only if the poset dimension of P (G)
is no more than three.
Thus our first step is to encode the linear extensions and the poset dimension of a poset P in terms of
polynomial equations. The idea is similar to our characterization of cycles via permutations.
Lemma 2.6. The poset P = (E, >) has poset dimension at most p if and only if the following system of
equations has a solution:
For k = 1, . . . , p :
|E|
Y
µ
(xi (k) − s) = 0,
for every i ∈ {1, . . . , |E|},
and
Y
sk
s=1
¶
xi (k) − xj (k) = 1 .
(2.4)
{i,j}∈{1,...,|E|},
i<j
For k = 1, . . . , p , and every ordered pair of comparable elements ei > ej in P :
xi (k) − xj (k) − ∆ij (k) = 0.
(2.5)
For every ordered pair of incomparable elements of P (i.e., ei 6> ej and ej 6> ei ) :
p
Y
¡
p
Y
¡
¢
xi (k) − xj (k) − ∆ij (k) = 0 ,
k=1
¢
xj (k) − xi (k) − ∆ji (k) = 0 ,
(2.6)
(∆ji (k) − d) = 0 .
(2.7)
k=1
For k = 1, . . . , p , and for every pair {i, j} ∈ {1, . . . , |E|}:
|E|−1
Y
|E|−1
Y
(∆ij (k) − d) = 0,
d=1
d=1
Proof. With Eqs. 2.4 and 2.5, we assign distinct numbers 1 through |E| to the poset elements, such that
the properties of a linear extension are satisfied. Eqs. 2.4 and 2.5 are repeated p times, so p linear extensions
are created. If the intersection of these extensions is indeed equal to the original poset P , then for every
incomparable pair of elements in P , at least one of the p linear extensions must detect the incomparability.
But this is indeed the case for Eq. 2.6, which says that for the l-th linear extension, the values assigned to
the incomparable pair ei , ej do not satisfy xi (l) < xj (l) , but instead satisfy xj (l) > xi (l) .
Proof (Theorem 1.1, Part 2). We simply apply the above lemma to the particular pairs of order relations
of the incidence poset of the graph. Note that in the formulation we added variables z{ij} that have the effect
of turning on or off an edge of the input graph.
Example 2.7 (Posets and Planar Graphs).
1
4
2
3
(1,2) (2,3)
1
2
(3,4)
3
(3,4)
(4,1)
(4,1) 4
(2,3)
3
4 (1,2)
2
1
(1,2)
(4,1)
1
(2,3)
2
(3,4)
3
4
(2,3)
(1,2)
2
(3,4)
3
(4,1)
4
1
Figure 2. Via Schnyder’s theorem, the square is planar since P (square) has dimension at most three.
8
J.A. De Loera, J. Lee, S. Margulies, S. Onn
Proof (Theorem 1.1, Part 3). Using Lemma 2.1, we can finish the proof of Part 3. For a k-colorable
subgraph H of size R , we set yij = 1 if edge {i, j} ∈ E(H) or yij = 0 otherwise. By Lemma 2.1, the resulting
subsystem of equations has a solution. Conversely from a solution, the subgraph H in question is read off
from those yij 6= 0 . Solvability implies that H is k-colorable.
Before we prove Theorem 1.1, Part 4, we recall that the edge-chromatic number of a graph is the minimum
number of colors necessary to color every edge of a graph such that no two edges of the same color are incident
on the same vertex.
Proof (Theorem 1.1, Part 4). If the system of equations has a solution, then Eq. 1.18 insures that
all variables xij are assigned ∆ roots of unity. Eq. 1.19 insures that no node is incident on two edges of
the same color. Since the graph contains a vertex of degree ∆ , the graph cannot have an edge-chromatic
number less than ∆ , and since the graph is edge-∆-colorable, this implies that the graph has edge-chromatic
number exactly ∆ . Conversely, if the graph has an edge-∆-coloring, simply map the coloring to the ∆ roots
of unity and all equations are satisfied. Since Vizing’s classic result shows that any graph with maximum
vertex degree ∆ can be edge-colored with at most ∆ + 1 colors, if there is no solution, then the graph must
have an edge-chromatic number of ∆ + 1 .
2.2. Normal forms and Dual colorings
In [2] Alon and Tarsi show another polynomial encoding of k-colorability. Here we consider one curious
consequence of the polynomial method for graph colorings when we use an algebraic encoding similar to that
of [2]. By taking a closer look at the normal form of the polynomials involved, we can derive a notion of dual
coloring, which has the nice property that a graph is dually d-colorable if and only if it is d-colorable. This
gives rise to an appealing new graph invariant: the simultaneous chromatic number σ(G) , defined to be the
infimal d such that G has a d-labeling that is simultaneously a coloring and a dual coloring.
¡ ¢
Fix a graph G = (V, E) with V := {1, . . . , n} and E ⊆ V2 , fix a positive integer d , and let D :=
0
d−1
{0, 1 . . . , d − 1} . Let α := exp( 2πi
d ) ∈ C be the primitive complex d-th root of unity, so that α , . . . , α
d
are distinct and α = 1 . For a d-labeling c : V −→ D of the vertices of G , let
o
Yn
²(c) :=
(αc(i) − αc(j) ) : i < j, {i, j} ∈ E .
Clearly, c is a proper d-coloring of G if and only if ²(c) 6= 0 .
With every orientation O = (V, A) of G (where A denotes the set of “arrows” or directed edges) associate
a sign signO = ±1 defined by the parity of the number |{(i, j) ∈ A : i > j}| of flips of O from the standard
orientation (where every directed edge (i, j) has i < j), and an out-degree vector δ O := (δ1O , . . . , δnO ) with δiO
the out-degree of vertex i in O . For a non-negative integer k let [k] ∈ D be the representative of k modulo
d , and for a vector δ = (δ1 , . . . , δn ) ∈ V n let [δ] = ([δ1 ], . . . , [δn ]) ∈ DV . For a labeling c∗ : V −→ D of the
vertices of G let
X©
ª
²∗ (c∗ ) :=
signO : O orientation of G with [δ O ] = c∗ .
Call c∗ a dual d-coloring of G if ²∗ (c∗ ) 6= 0 .
Theorem 2.8. A graph has a d-coloring, namely c ∈ DV with ²(c) 6= 0 (so is d-colorable) if and only if it
has a dual d-coloring, namely c∗ ∈ DV with ²∗ (c∗ ) 6= 0 (so is dually d-colorable).
Proof. Let G be a graph on n vertices. Consider the following radical zero-dimensional ideal I in C[x1 , . . . , xn ]
and its variety variety(I) in Cn :
I := hxd1 − 1, . . . , xdn − 1i ,
variety(I) := { αc := (αc(1) , . . . , αc(n) ) ∈ Cn : c ∈ DV } .
Expressing Combinatorial Problems by Polynomial Equations
9
It is easy to see that the set {xd1 − 1, . . . , xdn − 1} is a universal Gröbner basis (see [3] and references therein).
Qn
∗
∗
c∗ (i)
Thus, the (congruence classes of) monomials xc , c∗ ∈ DV (where xc := i=1 xi ), which are those
monomials not divisible by any xdi , form a vector space basis for the quotient C[x1 , . . . , xn ]/I . Therefore,
P
every polynomial f =
aδ ·xδ has a unique normal form [f ] with respect to this basis, namely the polynomial
∗
that lies in the vector space spanned by the monomials xc , c∗ ∈ DV , and satisfies f − [f ] ∈ I . It is not
P
very hard to show that this normal form is given by [f ] =
aδ · x[δ] .
Now consider the graph polynomial of G ,
Y
fG :=
{ (xi − xj ) : i < j, {i, j} ∈ E } .
The labeling c ∈ DV is a d-coloring of G if and only if ²(c) = fG (αc ) 6= 0 . Thus, G is not d-colorable if and
only if fG vanishes on every αc ∈ variety(I), which holds if and only if f ∈ I, since I is radical. It follows
P
O
that G is d-colorable if and only if the representative of fG is not zero. Since fG =
signO · xδ , with the
sum extending over the 2|E| orientations O of G , we obtain
X
X
O
∗
[fG ] =
signO · x[δ ] =
²∗ (c∗ ) · xc .
c∗ ∈D V
Therefore [fG ] 6= 0 and G is d-colorable if and only if there is a c∗ ∈ DV with ²∗ (c∗ ) 6= 0 .
Example 2.9. Consider the graph G = (V, E) having V = {1, 2, 3, 4} and E = {12, 13, 23, 24, 34} , and
let d = 3 . The normal form of the graph polynomial can be shown to be
[fG ]
=
x21 x22 x3 − x21 x22 x4 + x21 x2 x24 − x21 x2 x23 + x21 x23 x4 − x21 x3 x24 + x1 x2 − x1 x2 x23 x4 + x1 x23 x24
−x1 x3 + x1 x22 x3 x4 − x1 x22 x24 + x23 − x3 x4 + x22 x3 x24 − x22 + x2 x4 − x2 x23 x24 .
Note that in general, the number of monomials appearing in the expansion of fG can be as much as the
number of orientations 2|E| ; but usually it will be smaller due to cancellations that occur. Moreover, there will
usually be further cancellations when moving to the normal form, so typically [fG ] will have fewer monomials.
In our example, out of the 2|E| = 25 = 32 monomials corresponding to the orientations, in the expansion of
fG only 20 appear, and in the normal form [fG ] only 18 appear due to the additional cancellation:
−[x1 x33 x4 ] + [x1 x32 x4 ] = −x1 x4 + x1 x4 = 0 .
Note that the graph G in this example has only six 3-colorings (which are in fact the same up to relabeling
∗
of the colors), but as many as 18 dual 3-colorings c∗ corresponding to monomials xc appearing in [fG ] . For
instance, consider the labeling c∗ (1) = c∗ (2) = c∗ (4) = 0, c∗ (3) = 2: the only orientation O that satisfies
[δjO ] = c∗ (j) for all j is one with edges oriented as 21, 23, 24, 31, 34 , having signO = 1 and out-degrees δ1O =
Q4
c∗ (j)
= 1 · x01 x02 x23 x04 = x23 .
δ4O = 0 , δ3O = 2 and δ2O = 3 , contributing to [fG ] the non-zero term ²∗ (c∗ ) · j=1 xj
Thus, c∗ is a dual 3-coloring (but, since c∗ (1) = c∗ (2) , it is neither a usual 3-coloring nor a simultaneous
3-coloring — see below).
Note that in this example, and seemingly often, there are many more dual colorings than colorings; this
suggests a randomized heuristic to find a dual d-coloring for verifying d-colorability.
A particularly appealing notion that arises is the following: call a vertex labeling s : V −→ D a simultaneous
d-coloring of a graph G if it is simultaneously a d-coloring and a dual d-coloring of G . The simultaneous
chromatic number σ(G) is then the minimum d such that G has a simultaneous d-coloring. This is a strong
notion that may prove useful for inductive arguments, perhaps in the study of the 4-color problem of planar
graphs, and which provides an upper bound on the usual chromatic number χ(G) . First note that, like the
usual chromatic number, it can be bounded in terms of the maximum degree ∆(G) as follows.
Theorem 2.10. The simultaneous chromatic number of any graph G satisfies σ(G) ≤ ∆(G)+1 . Moreover,
10
J.A. De Loera, J. Lee, S. Margulies, S. Onn
for any G and d ≥ ∆(G) + 1 , there is an acyclic orientation O whose out-degree vector δ O = (δ1O , . . . , δnO )
provides a simultaneous d-coloring s defined by s(i) := δiO for every vertex i .
Proof. We prove the second (stronger) claim, by induction on the number n of vertices. For n = 1, this is
trivially true. Suppose n > 1 , and let d := ∆(G) + 1 . Pick any vertex i of maximum degree ∆(G) , and
let G0 be the graph obtained from G by removing vertex i and all edges incident on i . Let O0 be an acyclic
orientation of G0 and s0 the corresponding simultaneous d-coloring of G0 guaranteed to exist by induction.
Extend O0 to an orientation O of G by orienting all edges incident on i away from i , and extend s to the
corresponding vertex labeling of G by setting s(i) := δiO = d − 1 . Then O is acyclic, and therefore O is the
unique orientation of G with out-degree vector δ O . Thus,
X
²∗ (s) =
{ signθ : θ orientation of G with [δ θ ] = s = δ O } = ±1 6= 0 ,
and therefore s is a dual d-coloring of G . Moreover, if j is any neighbor of i in G , then the degree of j in
0
G0 is at most d − 2 , and therefore its label s0 (j) = δ O (j) ≤ d − 2 , and hence s(j) = s0 (j) 6= d − 1 = s(i) .
Therefore, s is also a d-coloring of G , completing the induction.
Example 2.11 (simultaneous 4-coloring of the Petersen graph).
According to Figure 3, δ O = (2, 1, 0, 2, 0, 3, 1, 2, 3, 1) . By inspection of Figure 3, s(i) := δiO does indeed
describe a valid 4-coloring of the Petersen graph.
1
2
6
2
3
5
8
9
10
7
4
3
1
0
0
2
3
1
1
2
Figure 3. Left: A vertex labeling. Right: An acyclic orientation labeled with out-degrees.
There are many fascinating new combinatorial and computational problems related to this new graph
invariant, the behavior of which is quite different from that of the usual chromatic number. For instance, the
direct analog of Brooks’ theorem, which states that every connected graph with maximum degree ∆ that is
neither complete nor an odd cycle is ∆-colorable, fails: It is not hard to verify that the simultaneous chromatic
number of the cycle Cn is 2 if and only if n is a multiple of 4; thus, the hexagon satisfies σ(C6 ) = 3 > ∆(C6 ) .
Which are the simultaneous chromatic Brooks graphs, i.e. those with σ(G) = ∆(G) ? What is the complexity
of deciding if a graph is simultaneously d-colorable? Which graphs are simultaneously d-colorable for small
d ? For d = 2 , the complete answer was given by L. Lovász [25] during a discussion at the Oberwolfach
Mathematical Institute:
Theorem 2.12. (Lovász) A connected bipartite graph G = (A, B, E) has simultaneous chromatic number
σ(G) = 2 if and only if at least one of |A| and |B| has the same parity as |E| .
3. Nullstellensatz Degree Growth in Combinatorics
The Hilbert Nullstellensatz states that a system of polynomial equations {f1 (x) = 0, f2 (x) = 0, . . . , fr (x) = 0} ⊆
C[x1 , . . . , xn ] has no solution in Cn if and only if there exist polynomials α1 , . . . , αr ∈ C[x1 , . . . , xn ] such
Expressing Combinatorial Problems by Polynomial Equations
11
P
that 1 =
αi fi (see [7]). The purpose of this section is to investigate the degree growth of the coefficients
αi . In particular, systems of polynomials coming from combinatorial optimization.
In our investigations, we will often need to find explicit Nullstellensatz certificates for specific graphs. This
can be done via linear algebra. First, given a system of polynomial equations, fix a tentative degree for the
coefficient polynomials αi in the Nullstellensatz certificates. This yields a linear system of equations whose
variables are the coefficients of the monomials of the polynomials α1 , . . . , αr . Then, solve this linear system.
If the system has a solution, we have found a Nullstellensatz certificate. Otherwise, try a higher degree for the
polynomials αi . For the Nullstellensatz certificates, the degrees of the polynomials αi cannot be more than
known bounds (see e.g., [18] and references therein), thus this is a finite (but potentially long) procedure to
decide whether a system of polynomials is feasible or not. In practice, sometimes low degrees suffice to find
a certificate.
Example 3.1. Suppose we wish to test K4 for 3-colorability, and we assume that the αi in the Nullstellensatz certificate have degree 1. After encoding K4 with the system of polynomial equations, we “conjecture”
that there exists a Nullstellensatz certificate of the following form
1 = (c1 x1 + c2 x2 + c3 x3 + c4 x4 + c5 )(x31 − 1) + (c6 x1 + c7 x2 + c8 x3 + c9 x4 + c10 )(x32 − 1)
+ (c11 x1 + · · · + c15 )(x33 − 1) + (c16 x1 + · · · + c20 )(x34 − 1)
+ (c21 x1 + · · · + c25 )(x21 + x1 x2 + x22 ) + (c26 x1 + · · · + c30 )(x21 + x1 x3 + x23 )
+ (c31 x1 + · · · + c35 )(x21 + x1 x4 + x24 ) + (c36 x4 + · · · + c40 )(x22 + x2 x3 + x23 )
+ (c41 x1 + · · · + c45 )(x22 + x2 x4 + x24 ) + (c46 x1 + · · · + c50 )(x23 + x3 x4 + x24 ).
When we multiply out this certificate, we group together like powers of x1 , x2 , x3 , x4 as follows:
1 = c1 x41 + · · · + c13 x43 + · · · + c8 x32 x3 + · · · + (c22 + c21 + c27 + c32 )x21 x2 + · · ·
+ (c35 + c45 + c50 )x24 + · · · + (−c15 − c20 − c5 − c10 ).
Because the Nullstellensatz certificate is identically 1, this identity gives rise to the following system of linear
equations: 0 = c1 , 0 = c13 , 0 = c8 , 0 = c22 + c21 + c27 + c32 , 0 = c35 + c45 + c50 , . . . , 1 = −c15 − c20 − c5 − c10 .
In other words, we have a large-scale sparse system of linear equations that consists only of 1s and −1s.
We implemented an exact-arithmetic linear system solver. In this example, it turns out that degree 1 is not
sufficient for generating a Nullstellensatz certificate — that is, this linear system has no solution. Ultimately,
we discovered that degree four is required, and we were able to produce the following certificate:
µ
¶
4 4 5 3
2 3
4 3
2 2
2 2
1=
−
− 1) +
x4 − x4 x2 − x4 x3 − x4 x1 + x4 x2 x1 + x4 x3 x1 (x24 + x2 x4 + x22 )
9
9
9
9
9
9
µ
¶
µ
¶
1 4 2 3
1 3
2 2
2
1
1
2
+
x4 + x4 x2 − x4 x1 − x4 x2 x1 (x22 + x3 x2 + x23 ) +
x44 + x34 x2 + x34 x1 + x24 x2 x1 (x24 + x3 x4 + x23 )
9
9
9
9
9
9
9
9
µ
¶
µ
¶
2
1
1
1
+ − x44 + x34 x1 − x4 x31 + x41 (x24 + x1 x4 + x21 ) + x34 x2 (x22 + x1 x2 + x21 ) + − x44 − x34 x2 (x23 + x1 x3 + x21 ).
3
3
3
3
(−x31
1)(x31
Lemma 3.2. If P 6= NP , then there must exist an infinite family of graphs whose minimum-degree non3-colorability Nullstellensatz certificates have unbounded growth with respect to the number of vertices and
edges in the graph.
Proof. Our proof is by contradiction with the hypothesis P 6= NP . Consider a non-3-colorable graph that
has been encoded as the system of polynomial equations (x3i − 1) = 0 for i ∈ V (G) , and (x2i + xi xj + x2j ) = 0
for {i, j} ∈ E(G) . Assume that every minimum-degree non-3-colorability Nullstellensatz certificate has
deg(αi ) < d for some constant d . We will show that P = NP by providing a polynomial-time algorithm
for solving Graph-3-Coloring: (1) Given a graph G , encode it as the above system of polynomial equations,
(2) Construct and solve the associated linear system for monomials of degree < d , (3) If the system has a
12
J.A. De Loera, J. Lee, S. Margulies, S. Onn
solution, a Nullstellensatz certificate exists, and the graph is non-3-colorable: Return no, (4) If the system
does not have a solution, there does not exist a Nullstellensatz certificate, and the graph is 3-colorable:
Return yes.
Now we analyze the running time of this algorithm. In Step 1, our encoding has one polynomial equation
per vertex and one polynomial equation per edge. Since there are O(n2 ) edges in a graph, our polynomial
system has n+n2 = O(n2 ) equations. Since every equation only contains coefficients ±1 and is of degree three
or less, encoding the graph as the above system of polynomial equations clearly runs in polynomial-time.
For Step 2, we note that by Corollary 3.2b of [34], if a system of linear equations Ax = b has a solution,
then it has a solution polynomially-bounded by the bit-sizes of the matrix A and the vector b (see [34] for a
definition of bit-size). In this case, the vector b contains only zeros and ones. To calculate the bit-size of A ,
we recall our assumption that, for every αi , deg(αi ) < d for some constant d . Therefore, an upper bound
on the number of terms in each αi is the total number of monomials in n variables of degree less than or
equal to d . Therefore, the number of terms in each αi is
µ
¶ µ
¶
µ
¶
n+d−1
n+d−2
n−1
+
+ ··· +
= O(nd ) + O(nd−1 ) + · · · + O(1) = O(nd ).
n−1
n−1
n−1
Since there are O(n2 ) equations, there are at most O(nd+2 ) unknowns in the linear system, and thus,
O(nd+2 ) columns in A . Since the vertex equations (x3i − 1) = 0 have two terms, and the edge equations
(x2i + xi xj + x2j ) = 0 have 3 terms, there are O(nd+2 ) terms in the expanded Nullstellensatz certificate, and
O(nd+2 ) rows in A . Since entries in A are 0, ±1 , the matrix A contains only entries of bit-size at most 2.
Therefore, the bit-sizes of both A and b are polynomially-bounded in n , and by Theorem 3.3 of [34], the
linear system can be solved in polynomial-time.
Therefore, we have demonstrated a polynomial-time algorithm for solving Graph-3-Coloring, and since
Graph-3-Coloring is NP-Complete ([12]), this implies P = NP , which contradicts our hypothesis. Therefore,
deg(αi ) d for any constant d .
Thus, in the linear algebra approach to finding a minimum-degree Nullstellensatz certificate, the existence
of a universal constant bounding the degree is impossible under a well-known hypothesis of complexity theory.
Clearly, a similar result can be obtained for other encodings (for a generalized statement see [26]). Note that
the linear algebra method does not rely on any property that is unique to a particular combinatorial or NPComplete problem; the only assumption is that the problem can be represented as a system of polynomial
equations. We will use it to find Nullstellensatz certificates of non-3-colorability and sizes of stable sets of
graphs.
We remark that this linear algebra method finds not only a Nullstellensatz certificate (if it exists), but
it finds one of minimum possible degree. With our implementation, we ran several experiments. We quickly
found out that the systems of linear equations are numerically unstable, thus it is best to use exact arithmetic
to solve them. The systems of linear equations are also quite large in practice, as the bound on the degree
of the polynomial coefficients grows. Thus we need ways to reduce the number of unknowns.
We will not discuss here ad hoc methods that depend on the particular polynomial system at hand (see [26]
for methods specific to 3-colorability), but one rather useful general trick is to randomly eliminate variables
in the above procedure. Instead of allowing all monomials of degree ≤ d to appear in the construction of the
linear system of equations, we can randomly set unknowns in the linear system of equations to 0 — e.g., set
each variable to 0 with probability p , independently, to get a smaller system.
In Figure 4, we see the results of the probabilistic search for smaller Nullstellensatz certificates. On the
x-axis is the probability p of keeping an unknown in the linear system. Thus, if p = 0.1 , 90% of the time we
set the unknown to 0, and only 10% of the time, we keep it in the system. For the cliques and odd wheels, we
know that there is always a certificate of degree four. For every probability 0.1, 0.2, . . . , 1 we performed 100
searches for a degree four certificate. For both the cliques and the odd wheels, at p = 0.1 and p = 0.2 , we
almost never found a certificate. But for p = 0.4 , we found certificates 95% of the time. In practice this idea
Expressing Combinatorial Problems by Polynomial Equations
13
Cliques
# of successes
100
K4
K5
K6
K7
80
60
40
20
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Odd Wheels
# of successes
100
80
3
5
7
60
40
20
0
0
0.1
0.2
0.3
0.4
0.5
probability
0.6
0.7
0.8
0.9
1
Figure 4. Probability tests on cliques and odd wheels.
is useful. We can reduce the number of variables in the linear system by 60%, and still find a Nullstellensatz
certificate 90% of the time. We will report the results of specific computations in Section 3.2.
3.1. The Nullstellensatz and Stable Sets of Graphs
Lovász [24] stated the challenge of finding an explicit family of graphs with growth in the degree of Nullstellensatz certificates. Here we solve his challenge. Our main result is stated in Theorem 1.2: For every graph
G , there exists a Nullstellensatz certificate of degree α(G) , the stability number of G , certifying that G
has no stable set of size > α(G) . Moreover, this is the minimum possible degree for all graphs. In what
follows, for any graph G with stability number α(G) and integer r ≥ 1 , we associate the infeasible system
of polynomial equations J(G, r) :
−(α(G) + r) +
n
X
xi = 0 ,
(3.1)
i=1
x2i − xi = 0, for every node i ∈ V (G) ,
(3.2)
xi xj = 0, for every edge {i, j} ∈ E(G) .
(3.3)
Thus, the Nullstellensatz certificate will have the general form:
µ
¶
n
X
X
1 = A − (α(G) + r) +
xi +
Qi (x2i − xi ) +
i=1
i∈V (G)
X
Qij (xi xj ) .
(3.4)
{i,j}∈E(G)
In the rest of this section, we will refer to the coefficient polynomials using these particular letters (i.e. A, Qi ,
etc).
Definition 3.3.
A Nullstellensatz certificate (Eq. 3.4) has degree d if max{deg(A), deg(Qi ), deg(Qij )} = d .
14
J.A. De Loera, J. Lee, S. Margulies, S. Onn
Lemma 3.4.
For any graph G and a Nullstellensatz certificate
µ
¶
n
X
X
xi +
Qi (x2i − xi ) +
1 = A − (α(G) + r) +
|
{z
i=1
}
i∈V (G)
X
Qij (xi xj )
(3.5)
{i,j}∈E(G)
B
certifying that G has no stable set of size (α(G)+r) (with r ≥ 1), we can construct a “reduced” Nullstellensatz
certificate
¶
µ
n
X
X
X
xi +
Q0i (x2i − xi ) +
Q0ij (xi xj ),
1 = A0 − (α(G) + r) +
i=1
i∈V (G)
{i,j}∈E(G)
such that
Pn
1. The coefficient A0 multiplying −(α(G) + r) + i=1 xi has only square-free monomials supported on stable
sets of G , and thus deg(A0 ) ≤ α(G) .
2. max{deg(A), deg(Qi ), deg(Qij )} = max{deg(A0 ), deg(Q0i ), deg(Q0ij )} . Thus, if the original Nullstellensatz
certificate has minimum-degree, the “reduced” certificate also has minimum-degree.
Proof. Let I be the ideal generated by x2i − xi (for every node i ∈ V (G)), and xi xj ( for every edge {i, j} ∈
E(G)). We apply reductions modulo I to Eq. 3.5. If a non-square-free monomial appears in polynomial A , say
αj −2 αk
αk
1 α2
1 α2
xα
we can subtract the polynomial xα
xik B(x2ij −
i1 xi2 · · · xik with at least one αj > 1 , thenP
i1 xi2 · · · , xij
2
xij ) from AB and simultaneously add it to
Qs (xs − xs ) . Thus, eventually we obtain a new certificate
that has only square-free monomials in A0 . Furthermore, if Q0s has new monomials, they are of degree less
than or equal to what was originally in A .
Similarly, if xi1 xi2 · · · xik appears in A , but xi1 xi2 · · · xik contains an edge {i, j} ∈ E(G) (if xi xj divides
xi1 xi2 · · · xik ), then we can again subtract B(xi1 xi2 · · · xik /xi xj )(xi xj ) from AB , and, at the same time, add
P
it to {i,j}∈E(G) Qij xi xj . Furthermore, the degree is maintained, and we have reached the form we claim
exists for A0 .
We now show that, for every graph, there exists an explicit Nullstellensatz certificate of degree α(G) .
Theorem 3.5. Given a graph G , there exists a Nullstellensatz certificate of degree α(G) certifying the
non-existence of stable sets of size greater than α(G) .
Proof. The proof is an algorithm to construct the explicit Nullstellensatz certificate for the non-existence
of a stable set of size α(G) + r, with r ≥ 1 . First, let us establish some notation: Let S(i, G) be the set
of all stable sets of size i in G . We index nodes in the graph by integers, thus stable sets in G are subsets
of integers. When we refer to a monomial xd1 xd2 · · · xdi as a “stable set”, we mean {d1 , . . . , di } ∈ S(i, G) .
We use “hat” notation to remove a variable from a monomial, meaning xd1 xd2 · · · xdi−1 xc
di xdi+1 · · · xdk =
xd1 xd2 · · · xdi−1 xdi+1 · · · xdk . Finally, let
X
P(i, G) :=
xd1 xd2 · · · xdi ,
and P(0, G) := 1 .
{d1 ,d2 ,...,di }∈S(i,G)
When P(i, G) is ordered lexicographically, we denote Pj (i, G) as the j-th term. We also define the constants
CiG :=
G
iCi−1
,
α(G) + r − i
and
C0G :=
1
.
α(G) + r
Our main claim is that we can construct explicit coefficients A, Qi , Qij of degree less than or equal to α(G)
Expressing Combinatorial Problems by Polynomial Equations
15
********************************************
ALGORITHM (Nullstellensatz certificate construction)
INPUT: A graph G(V, E), associated polynomials P(i, G) , α(G), r
OUTPUT: polynomials (A, Qi , Qij ) such that AB + C = 1 is true
P
G
0 A ← − α(G)
i=0 Ci P(i, G)
1 Qi ← 0 , for i = 1 . . . |V |
2 Qij ← 0 , for {i, j} ∈ E(G)
3 for i ← 0 to α(G)
4
for j ← 1 to # of monomials in P(i, G)
5
for k ← 1 to |V |
6
Let Pj (i, G) = xd1 xd2 · · · xdi
7
if Pj (i, G)xk is a square-free stable set
G
8
(rule 1) Qk ← Qk + Ci+1
xd1 xd2 · · · xdi
9
else if Pj (i, G)xk is a square-free non-stable set
10
Choose an edge l, 1 ≤ l ≤ i, so that (k, dl ) is an edge of G
12
(rule 2) Qkdl ← Qkdl + CiG xd1 xd2 · · · xdl−1 xc
dl xdl+1 · · · xdi
13
end if
14
end for
15
end for
16 end for
17 return A, Qi , Qij
*******************************************
Figure 5. Nullstellensatz certificate construction
such that the following identity is satisfied
µ
¶
n
X
1 = A − (α(G) + r) +
xi +
|
{z
B
i=1
}
X
{i,j}∈E(G)
|
Qij xi xj +
{z
n
X
i=1
Qi (x2i − xi ) .
}
C
We do so by the algorithm of Figure 5. In Figure 5 and in what follows, AB and C refer to parts of the
equation as marked above.
It is clear that the algorithm terminates for any finite graph, since the number of iterations of each for
loop is finite. To prove correctness of the algorithm, we demonstrate that the following statement is true for
each iteration of the for loop beginning in line 3:
Prior to the m-th iteration, AB + C − 1 only contains terms of degree m + 1 or greater.
(∗)
Furthermore, all terms of degree m + 1 are square-free.
Pα(G)
When the for loop is initialized, A is set to the polynomial − i=0 CiG P(i, G) , and Qi , Qij are set to
zero. Therefore, prior to the 0-th iteration, C = 0 and AB + C − 1 = AB − 1 . Since the constant term in
AB is equal to
µ
¶
¡
¢
1
G
−C0 P(0, G) − (α(G) + r) = −
− (α(G) + r) = 1 ,
α(G) + r
this implies that AB −1 only contains terms of degree 1 or higher. Furthermore, linear terms are by definition
square-free. Thus, (∗) is true at initialization.
Now we must show that if (∗) is true prior to the m-th iteration of the for loop, then it will be true prior
to the (m + 1)-th iteration. Assume then, that prior to the m-th iteration, AB + C − 1 only contains terms
of degree m + 1 or greater, and that all terms of degree m + 1 are square-free. We must show that prior to
the (m + 1)-th iteration, AB + C − 1 only contains terms of degree m + 2 or greater, and furthermore, that
all terms of degree m + 2 are square-free. Prior to the m-th iteration, there are only two kinds of terms of
16
J.A. De Loera, J. Lee, S. Margulies, S. Onn
degree m + 1 in AB + C − 1 : (1) terms corresponding to square-free, stable sets, and (2) terms corresponding
to square-free, non-stable sets. We will show that both kinds of terms cancel during the m-th iteration.
• Let xd1 xd2 · · · xdm+1 be any square-free, stable set monomial in AB + C − 1 of degree (m + 1) . Since
{d1 , d2 , . . . , dm+1 } is a stable set, all subsets of size m are likewise stable sets and appear as summands
in P(m, G) . Consider the coefficient
¡
¢ of xd1 xd2 · · · xdm+1 in C . During the m-th iteration, we apply rule
1 and create this monomial m+1
times in C . Since this monomial is created by the multiplication of
m
2
xd1 xd2 · · · x
d
·
·
·
x
with
(x
−
x
dk ) , the coefficient for xd1 xd2 · · · xdm+1 in C is
dk
dm+1
dk
µ
¶
m+1
G
G
−
Cm+1
= −(m + 1)Cm+1
m
Now we will calculate the coefficient of this same monomial in AB . This monomial is created in two
G
ways, (1) multiplying −Cm
xd1 xd2 · · · xdj−1 xc
dj xdj+1 · · · xdm+1 by xdj (repeated (m + 1) times, once for each
G
xdj ), or (2) multiplying −Cm+1
xd1 xd2 · · · xdm+1 by −(α(G) + r) (occurring exactly once). Therefore, the
coefficient of xd1 xd2 · · · xdm+1 in AB is
¡
¢
G
G
− Cm+1
− (α(G) + r) − (m + 1)Cm
G
(m + 1)Cm
(α(G) + r − (m + 1))
α(G) + r − (m + 1)
¡
¢
G
G
G
+ rCm+1
− Cm+1
α(G) + r − (m + 1)
= α(G)Cm+1
G
G
+ rCm+1
−
= α(G)Cm+1
G
G
G
G
G
= α(G)Cm+1
+ rCm+1
− α(G)Cm+1
− rCm+1
+ (m + 1)Cm+1
G
= (m + 1)Cm+1
.
Therefore, the coefficient for any square-free stable-set monomial in AB + C − 1 is
G
G
(m + 1)Cm+1
−(m + 1)Cm+1
=0.
|
{z
}|
{z
}
from AB
from C
• Now, consider¡ any¢ square-free non-stable-set monomial xd1 xd2 · · · xdm+1 in AB + C − 1 of degree m + 1 .
Consider
all m+1
subsets of {d1 , d2 , . . . , dm+1 } , and let M be the number of stable sets among those
m
¡m+1¢
subsets.
Each
of those M subsets appears as a summand in P(m, G) . Therefore, the monomial
m
xd1 xd2 · · · xdm+1 is created M times in AB , and M times in C , by M applications of rule 2 . Therefore,
the coefficient for any square-free non-stable set monomial xd1 xd2 · · · xdm+1 in AB + C − 1 is
−M C G + M C G = 0 .
| {z m} | {z m}
from AB
from C
Finally, consider any non-square-free monomial xd1 xd2 · · · x2dl · · · xdm+1 in AB + C − 1 of degree m + 2 .
We note that {d1 , d2 , . . . , dm+1 } is an stable set. To see this, note that every non-square-free monomial
in AB is created by the product of an stable set with a linear term, and every non-square-free monomial
in C is created by the product of an stable set with (x2dl − xdl ) for some dl . During the m-th iteration,
during applications of rule 1, xd1 xd2 · · · xc
dl · · · xdm+1 is added to Qdl . Therefore, when Qdl is subsequently
2
multiplied by (xdl − xdl ) , the monomial xd1 xd2 · · · x2dl · · · xdm+1 is created. To summarize, this monomial is
created in only one way in AB and only one way in C . Therefore, the coefficient for xd1 xd2 · · · x2dl · · · xdm+1
in AB + C − 1 is
G
G
−Cm+1
+ Cm+1
=0.
| {z } | {z }
from AB
from C
Therefore, we have proven that (∗) is valid prior to every iteration of the for loop. Finally, upon termination, i = α(G) + 1 . Therefore, by (∗), we know that AB + C − 1 only contains monomials of degree α(G) + 2
or greater. However, there are no terms of degree α(G) + 2 in AB since deg(AB) = α(G) + 1 . Additionally,
during the α(G)-th iteration, P(α(G), G)xk is never an stable set. Therefore, only applications of rule 2
occur during the last iteration, and the final degrees of Qi , Qij are less than or equal to α(G) − 1 . Therefore,
Expressing Combinatorial Problems by Polynomial Equations
17
the monomial in C of greatest degree is of degree α(G) + 1 . Thus, there are no monomials in AB + C − 1
of degree α(G) + 2 or greater, and upon termination, AB + C − 1 = 0 .
We have then shown that we can construct A, Qi , Qij such that
µ
1=
|
¶µ
α(G)
−
X
i=0
C(i, G)P(i, G)
{z
− (α(G) + r) +
}|
i=1
{z
A
n
X
¶
xi +
X
Qij xi xj +
{i,j}∈E(G)
}
|
B
{z
n
X
Qi (x2i − xi ) .
i=1
}
C
Since deg(Qi ), deg(Qij ) ≤ (α(G) − 1) , and deg(A) = α(G) , this concludes our proof.
Example 3.6. We display a certificate found by our algorithm. In Figure 6 is the Turán graph T (5, 3) .
It is clear that α(T (5, 3)) = 2 . Therefore, we “test” for a stable set of size 3.
3 4
2
1
5
Figure 6. Turán graph T (5, 3)
The certificate constructed by our algorithm is
µ
¶
¢ 1¡
¢ 1
1¡
1 = − x1 x2 + x3 x4 − x1 + x2 + x3 + x4 + x5 −
(x1 + x2 + x3 + x4 + x5 − 3)+
3
6
3
¶
µ
¶
µ
¶
µ
¶
µ
1
1
1
1
1
1
1
1
1
x4 + x2 +
x1 x3 +
x2 +
x1 x4 +
x2 +
x1 x5 +
x4 +
x2 x3 +
3
3
3
3
3
3
3
3
3
µ ¶
µ ¶
µ
¶
µ ¶
µ
¶
1
1
1
1
1
1
1
x2 x4 +
x2 x5 +
x4 +
x3 x5 +
x4 x5 +
x2 +
(x21 − x1 )+
3
3
3
3
3
3
6
¶
µ
¶
µ
¶
µ ¶
µ
1
1
1
1
1
1
1
x1 +
(x22 − x2 ) +
x4 +
(x23 − x3 ) +
x3 +
(x24 − x4 ) +
(x25 − x5 ) .
3
6
3
6
3
6
6
Note that the coefficient for the stable set polynomial contains every stable set, and further note that
every monomial in every coefficient is also indeed a stable set.
We will now prove that the stability number α(G) is the minimum-degree for any Nullstellensatz certificate
for the non-existence of a stable set of size greater than α(G) . To prove this, we rely on two propositions.
Proposition 3.7.
Given a graph G , let M = {d1 , d2 , . . . , d|M | } be any maximal stable set in G . Let
µ
¶
n
n
X
X
X
1 = A − (α(G) + r) +
xi +
Qij xi xj +
Qi (x2i − xi )
i=1
i=1
{i,j}∈E(G)
be a Nullstellensatz certificate for the non-existence of a stable set of size α(G) + r (with r ≥ 1), and let
µ
¶
n
n
X
X
X
Q0i (x2i − xi )
Q0ij xi xj +
(3.6)
1 = A0 − (α(G) + r) +
xi +
|
{z
B
i=1
}
{i,j}∈E(G)
|
{z
C
}
i=1
|
{z
}
D
be the reduced certificate via Lemma 3.4. Then, for i ∈ {1, . . . , |M |} , the linear term xdi appears in A0 with
a non-zero coefficient.
18
J.A. De Loera, J. Lee, S. Margulies, S. Onn
Proof. Our proof is by contradiction. Assume that xdi does not appear in A0 with a non-zero coefficient.
By inspection of Eq. 3.6, we see that A0 must contain the constant term a0 = −(α(G) + r) . Therefore, the
term a0 xdi appears in A0 B . However, a0 xdi does not cancel within A0 B since xdi does not appear in A0 by
assumption. Therefore, a0 xdi must cancel with a term elsewhere in the certificate. Specifically, since C only
contains terms multiplying edge monomials, a0 xdi must cancel with a term in D . But the linear term a0 xdi
is generated in D in only one way: a0 must multiply (x2di − xdi ) Therefore,
Q0di = other terms + a0 .
But when a0 multiplies (x2di − xdi ) , this not only generates −a0 xdi , which neatly cancels its counterpart in
A0 B , but it also generates the cross-term a0 x2di , which must cancel elsewhere in the certificate. However,
x2di does not cancel with a term C , since C contains only terms multiplying edge monomials, and x2di does
not cancel with a term in A0 B , since A0 contains only terms corresponding to square-free stable sets and
also because A0 does not contain xdi by assumption. Therefore, a0 x2di must cancel elsewhere in D . There is
only one way to generate a second x2di term in D: a0 xdi must multiply (x2di − xdi ) . Therefore,
Q0di = other terms + a0 xdi + a0 .
Now, we assume
2
Q0di = other terms + a0 xkdi + a0 xk−1
di · · · + a0 xdi + a0 xdi + a0 .
When a0 xkdi multiplies (x2di − xdi ) , this generates a cross term of the form a0 xk+2
. This term must cancel
di
k+2
elsewhere in the certificate. As before, a0 xdi does not cancel with a term in C , since C contains only terms
multiplying edge monomials, and a0 xk+2
does not cancel with a term in A0 B , since A0 contains only terms
di
corresponding to square-free stable sets. Therefore, a0 xk+2
must cancel elsewhere in D . But as before, there
di
k+1
is only one way to generate a second a0 xk+2
term
in
D:
a
x
must multiply (x2di − xdi ) . Therefore,
0 di
di
k
2
Q0di = other terms + a0 xk+1
di + a0 xdi + · · · + a0 xdi + a0 xdi + a0 .
To summarize, we have inductively shown that in order to cancel lower-order terms, we are forced to generate
terms of higher and higher degree. In other words, Qdi contains an infinite chain of monomials increasing
in degree. Since deg(Q0di ) is finite, this is clearly a contradiction.
Therefore, xdi must appear in A0 with a non-zero coefficient.
Proposition 3.8. Given a graph G , let M = {d1 , d2 , . . . , d|M | } be any maximal stable set in G , and let
{c1 , c2 , . . . , ck+1 } be any (k + 1)-subset of M with k < |M | . Let
µ
¶
n
n
X
X
X
1 = A − (α(G) + r) +
xi +
Qij xi xj +
Qi (x2i − xi )
i=1
i=1
{i,j}∈E(G)
be a Nullstellensatz certificate for the non-existence of a stable set of size α(G) + r (with r ≥ 1), and let
µ
¶
n
n
X
X
X
1 = A0 − (α(G) + r) +
xi +
Q0ij xi xj +
Q0i (x2i − xi )
|
{z
B
i=1
}
{i,j}∈E(G)
|
{z
C
}
i=1
|
{z
}
D
be the reduced certificate via Lemma 3.4. If xc1 xc2 · · · xck appears in A0 with a non-zero coefficient, then
xc1 xc2 · · · xck xck+1 also appears in A0 with a non-zero coefficient.
Proof. Our proof is by contradiction. Assume xc1 xc2 · · · xck with k < |M | appears in A0 with a non-zero
coefficient, but xc1 xc2 · · · xck xck+1 does not. Since xc1 xc2 · · · xck appears in A0 , xc1 xc2 · · · xck xck+1 clearly
appears in A0 B and must cancel elsewhere in the certificate. However, xc1 xc2 · · · xck xck+1 does not cancel
with a term in C , since C contains only terms multiplying edge monomials and {c1 , c2 , . . . , ck+1 } is a
Expressing Combinatorial Problems by Polynomial Equations
19
stable set. Furthermore, xc1 xc2 · · · xck xck+1 does not cancel with a term in A0 B , since A0 does not contain
xc1 xc2 · · · xck xck+1 by assumption. Therefore, xc1 xc2 · · · xck xck+1 must cancel with a term in D , and for at
least one i , xc1 xc2 · · · xc
ci · · · xck xck+1 appears in Qci with a non-zero coefficient.
2
2
When xc1 · · · xc
ci · · · xck xck+1 multiplies (xci −xci ) , this generates a cross term of the form xc1 · · · xci · · · xck xck+1
2
which must cancel elsewhere in the certificate. Let m1 = xc1 xc2 · · · xck xck+1 and m2 = xc1 xc2 · · · xci · · · xck xck+1 .
Note that deg(m2 ) = deg(m1 ) + 1 . As before, m2 does not cancel with a term in C , since C contains only
terms multiplying edge monomials, and m2 does not cancel with a term in A0 B , since A0 contains only
terms corresponding to square-free stable sets. Therefore, m2 must cancel elsewhere in D .
In order to cancel m2 in D , for some cj , we must subtract one from the cj -th exponent, and then multiply
this monomial by (x2cj − xcj ) . However, this generates a cross-term m3 = xc1 xc2 · · · x2ci · · · x2cj · · · xck xck+1
where deg(m3 ) = deg(m2 ) + 1 . Note that in the case when j = i , m3 = xc1 xc2 · · · x3ci · · · xck xck+1 , but
deg(m3 ) still is equal to deg(m2 ) + 1 .
Inductively, consider the n-th element in this chain, and assume it appears with a non-zero coefficient in
some Qcj . Let
αk+1
1 α2
mn = xα
c1 xc2 · · · xck+1 ,
where αi ≥ 1 for i ∈ {1, . . . , k + 1} . When mn multiplies (x2cj − xcj ) , this generates the cross-term mn x2cj .
This term must cancel elsewhere in the certificate. As before, mn x2cj does not cancel with a term C , since
C contains only terms multiplying edge monomials, and mn x2cj does not cancel with a term in A0 B , since
A0 contains only terms corresponding to square-free stable sets and xc1 xc2 · · · xck xck+1 does not appear in
A0 by assumption. Therefore, mn x2cj must cancel elsewhere in D .
In order to cancel mn x2cj in D , note that
αj +2
1 α2
k+1
mn cj 2 = xα
· · · xα
c1 xc2 · · · xcj
ck+1 ,
and for some l , let
αj +2
1 α2
l −1
k+1
mn+1 = xα
· · · xα
· · · xα
c1 xc2 · · · xcj
cl
ck+1 .
Note that deg(mn+1 ) = deg(mn ) + 1 . Therefore, in order to cancel mn x2cj , mn+1 multiplies (x2cl − xcl ) ,
which generates a new term of higher degree: mn+1 x2cl .
To summarize, we have inductively shown that in order to cancel lower-order terms, we are forced to
generate terms of higher and higher degree. In other words, m1 , m2 , . . . form an infinite chain of monomials
increasing in degree. Since deg(Q0i ) is finite, this is clearly a contradiction.
Therefore, xc1 xc2 · · · xck xck+1 must appear in A0 with a non-zero coefficient.
Using Propositions 3.7 and 3.8, we can now prove the main theorem of this section.
Theorem 3.9. Given a graph G , any Nullstellensatz certificate for the non-existence of a stable set of size
greater than α(G) has degree at least α(G) .
Proof.
Our proof is by contradiction. Let
µ
¶
n
X
1 = A − (α(G) + r) +
xi +
i=1
X
Qij xi xj +
n
X
Qi (x2i − xi )
i=1
{i,j}∈E(G)
be any Nullstellensatz certificate for the non-existence of a stable set of size α(G) + r, with r ≥ 1 , such that
deg(A), deg(Qi ), deg(Qij ) < α(G) , and let
µ
¶
n
n
X
X
X
0
0
Qij xi xj +
Q0i (x2i − xi )
(3.7)
1 = A − (α(G) + r) +
xi +
|
{z
B
i=1
}
{i,j}∈E(G)
|
{z
C
}
i=1
|
{z
D
}
20
J.A. De Loera, J. Lee, S. Margulies, S. Onn
be the reduced certificate via Lemma 3.4. The proof of Lemma 3.4 implies deg(A0 ) ≤ deg(A) < α(G) . Let
M = {d1 , d2 , . . . , dα(G) } be any maximum stable set in G . Via Proposition 3.7, we know that xd1 appears in
A0 with a non-zero coefficient, which implies (via Proposition 3.8) that xd1 xd2 appears in A0 with a non-zero
coefficient, which implies that xd1 xd2 xd3 appears in A0 and so on. In particular, xd1 xd2 · · · xdα(G) appears
in A0 . This contradicts our assumption that deg(A0 ) < α(G) . Therefore, there can be no Nullstellensatz
certificate with deg(A) < α(G) , and the degree of any Nullstellensatz certificate is at least α(G) .
Propositions 3.7 and 3.8 also give rise to the following corollary.
Corollary 3.10. Given a graph G , any Nullstellensatz certificate for the non-existence of a stable set of
size greater than α(G) contains at least one monomial for every stable set in G .
Proof. Given any Nullstellensatz certificate, we can create the reduced certificate via Lemma 3.4. The
proof of the Lemma 3.4 implies that the number of terms in A is equal to the number of terms in A0 . Via
Propositions 3.7 and 3.8, A0 contains one monomial for every stable set in G . Therefore, A also contains
one monomial for every stable set in G .
This brings us to the last theorem of this section.
Theorem 3.11. Given a graph G , a minimum-degree Nullstellensatz certificate for the non-existence of a
stable set of size greater than α(G) has degree equal to α(G) and contains at least one term for every stable
set in G .
Proof.
This theorem follows directly from Theorems 3.5, 3.9, and Corollary 3.10.
Finally, our results establish new lower bounds for the degree and number of terms of Nullstellensatz
certificates. In earlier work, researchers in logic and complexity showed both logarithmic and linear growth
of degree of the Nullstellensatz over finite fields or for special instances, e.g. Nullstellensatz related to the
pigeonhole principle (see [5], [16] and references therein). Our main complexity result below settles a question
of Lovász [24]:
Corollary 3.12. Given any infinite family of graphs Gn , on n vertices, the degree of a minimum-degree
Nullstellensatz certificate for the non-existence of a stable set of size greater than α(G) grows as Ω(n) .
Moreover, there are graphs for which the degree of the Nullstellensatz certificate grows linearly in n and,
at the same time, the number of terms in the coefficient polynomials of the Nullstellensatz certificate is
exponential in n .
Proof.
The stability number of a graph G with n nodes and m vertices grows linearly ([14]) since
µ
¶
p
1
α(G) ≥
(2m + n + 1) − (2m + n + 1)2 − 4n2 .
2
Finally, it is enough to remark that there exist families of graphs with linear growth in the minimum degree
of their Nullstellensatz certificates, but exponential growth in their numbers of terms. The disjoint union
of n/3 triangles has exactly 3n/3 maximal stable sets. Therefore, its Nullstellensatz certificate’s minimum
degree grows as O(n/3) , but its number of terms grows as 3n/3 (see [13] and references therein).
Expressing Combinatorial Problems by Polynomial Equations
21
3.2. The Nullstellensatz and 3-colorability
In this section, we investigate the degree growth of Nullstellensatz certificates for the non-3-colorability for
certain graphs. Curiously, every non-3-colorable graph that we have investigated so far has a minimum-degree
Nullstellensatz certificate of degree four. Next, we prove that four is indeed a lower bound on the degree of
such certificates.
Theorem 3.13.
Every Nullstellensatz certificate of a non-3-colorable graph has degree at least four.
Proof. Our proof is by contradiction. Suppose there exists a Nullstellensatz certificate of degree three or
less. Such a certificate has the following form
1=
n
X
P{i} (x3i − 1) +
i=1
X
P{ij} (x2i + xi xj + x2j ) ,
(3.8)
{i,j}∈E
where P{i} and P{ij} represent general polynomials of degree less than or equal to three. To be precise,
P{i} =
n
X
a{i}s x3s +
s=1
+
n X
n
X
s=1
n
n
X
X
n
X
b{i}st x2s xt
t=1
t6=s
c{i}stu xs xt xu +
s=1 t=s+1 u=t+1
n X
n
X
d{i}st xs xt +
s=1 t=1
n
X
e{i}s xs + f{i}
s=1
and
P{ij} =
n
X
a{ij}s x3s +
s=1
+
n
n
X
X
n X
n
X
s=1
n
X
b{ij}st x2s xt
t=1
t6=s
c{ij}stu xs xt xu +
s=1 t=s+1 u=t+1
n X
n
X
d{ij}st xs xt +
s=1 t=1
n
X
e{ij}s xs + f{ij} .
s=1
Since we work with undirected graphs, note that a{ij}s = a{ji}s , and this fact applies to all coefficients a
through f . Note also that when {i, j} is not an edge of the graph, Pij = 0 and thus a{ij}s = 0 . Again, this
fact holds for all coefficients a through f .
When P{i} multiplies (x3i −1) , this generates cross-terms of the form P{i} x3i and −P{i} . In particular, this
generates monomials of degree six or less. Notice that P{ij} (x2i + xi xj + x2j ) does not generate monomials of
degree six, only monomials of degree five or less. We begin the process of deriving a contradiction from Eq.
3.8 by considering all monomials of the form x3s x3i that appear in the expanded Nullstellensatz certificate.
These monomials are formed in only two ways: Either (1) x3s (x3i − 1) , or (2) x3i (x3s − 1) . Therefore, the n2
equations for x3s x3i are as follows:
0=
a{1}1 ,
0=
..
.
a{1}2 + a{2}1 ,
..
.
0=
a{n−1}n + a{n}(n−1) ,
0=
a{n}n .
(coefficient for x31 x31 = x61 )
(coefficient for
..
.
x31 x32 )
(coefficient for x3(n−1) x3n )
(coefficient for x3n x3n = x6n )
(I.1)
(I.2)
(I.n2 − 1)
(I.n2 )
Summing equations I.1 through I.n2 , we get
0=
n X
n
X
i=1 s=1
a{i}s .
(3.9)
22
J.A. De Loera, J. Lee, S. Margulies, S. Onn
Let us now consider monomials of the form x2s xt x3i (with s 6= t). These monomials are formed in only one
way: by multiplying b{i}st x2s xt by x3i . Therefore, since the coefficient for x2s xt x3i must simplify to zero in
the expanded Nullstellensatz certificate, b{i}st = 0 for all b{i} . When we consider monomials of the form
xs xt xu x3i (with s < t < u) , we see that c{i}stu = 0 for all c{i} , for the same reasons as above.
As we continue toward our contradiction, we now consider monomials of degree three in the expanded
Nullstellensatz certificate. In particular, we consider the coefficient for x3s . The monomial x3s is generated in
three ways: (1) f{s} (x3s −1) , (2) a{i}s x3s (x3i −1) (from the vertex polynomials), and (3) e{st}s xs (x2s +xs xt +x2t )
(from the edge polynomials). The equations for x31 , . . . , x3n are as follows:
0=
f{1} −
n
X
i=1
0=
f{2} −
n
X
X
a{i}1 +
e{1t}1 ,
(coefficient for x31 )
(II.1)
e{2t}2 ,
(coefficient for x32 )
(II.2)
t∈Adj(1)
X
a{i}2 +
i=1
t∈Adj(2)
..
.
..
.
0=
f{n} −
n
X
X
a{i}n +
i=1
..
.
e{nt}n .
(coefficient for x3n )
(II.n)
t∈Adj(n)
Summing equations II.1 through II.n , we get
0=
n
X
f{i} −
µX
n X
n
¶
a{i}s
i=1 s=1
i=1
+
n
X
n
X
e{st}s .
(3.10)
s=1 t∈Adj(s)
Since the degree three or less Nullstellensatz certificate (Eq. 3.8) is identically one, the constant terms must
Pn
sum to one. Therefore, we know i=1 f{i} = − 1 . Furthermore, recall that e{st}s = 0 if the undirected edge
{s, t} does not exist in the graph. Therefore, applying Eq. 3.9 to Eq. 3.10, we have the following equation
1=
n
X
e{st}s .
(3.11)
s,t=1,
s6=t
To give a preview of our overall proof strategy, the equations to come will ultimately show that the right-hand
side of Eq. 3.11 also equals zero, which is a contradiction.
Now we will consider the monomial x2s xt (with s 6= t). We recall that b{i}st = 0 for all b{i} (where b{i}st
is the coefficient for x2s xt in the i-th vertex polynomial). Therefore, we do not need to consider b{i}st in the
equation for the coefficient of monomial x2s xt . In other words, we only need to consider the edge polynomials,
¡ ¢
which can generate this monomial in two ways: (1) e{st}s xs · xs xt , and (2) e{si}t xt · x2s . The N = 2 n2
equations for these coefficients are:
0=
X
e{12}1 +
e{1i}2 ,
(coefficient for x21 x2 )
(III.1)
e{1i}3 ,
(coefficient for x21 x3 )
(III.2)
i∈Adj(1)
0=
X
e{13}1 +
i∈Adj(1)
..
.
0=
e{n(n−1)}n +
X
i∈Adj(n)
..
.
e{ni}(n−1) .
..
.
(coefficient for x2n xn−1 )
(III.N )
Expressing Combinatorial Problems by Polynomial Equations
When we sum equations III.1 to III.N , we obtain
µX
¶ µX
n X
n
n
n
X
X
e{st}s +
e{st}t +
s=1
t=1,
|
t6=s
s=1 t∈Adj(s)
{z
n
X
X
s=1 t∈Adj(s)
}
¶
=0.
e{st}u
(3.12)
u=1,
u6=s,t
{z
|
partial sum A
23
}
partial sum B
However, recall that e{st}u = 0 when {s, t} does not exist in the graph, and also that e{st}t = e{ts}t . Thus,
we can rewrite partial sum A from Eq. 3.12 as
n
X
X
e{st}t =
s=1 t∈Adj(s)
n X
n
X
s=1
e{st}t =
n X
n
X
s=1
t=1,
t6=s
e{ts}t .
t=1,
t6=s
Substituting the above into Eq. 3.12 yields
n
X
2
µX
n
e{st}s +
n
X
X
s=1 t∈Adj(s)
s,t=1,
s6=t
¶
u6=s,t
{z
|
=0.
e{st}u
(3.13)
u=1,
}
partial sum B
Finally, we consider the monomial xs xt xu (with s < t < u). We have already argued that c{i}stu = 0 for all
c{i} (where c{i}stu is the coefficient for xs xt xu in the i-th vertex polynomial). Therefore, as before, we need
only consider the edge polynomials, which can generate this monomial in three ways: (1) e{st}u xu · xs xt , (2)
e{su}t xt · xs xu , and (3) e{tu}s
¡n¢xs · xt xu . As before, these coefficients must cancel in the expanded certificate,
which yields the following 3 equations:
0=
e{12}3 + e{13}2 + e{23}1 ,
(coefficient for x1 x2 x3 )
(IV.1)
0=
..
.
e{12}4 + e{14}2 + e{24}1 ,
..
.
(coefficient for x1 x2 x4 )
..
.
(IV.2)
0=
e{(n−2)(n−1)}n + e{(n−2)n}(n−1) + e{(n−1)n}(n−2) .
(coefficient for xn−2 xn−1 xn )
Summing equations IV.1 through IV.M , we obtain
µ
¶
n−2
n
X n−1
X X
e{st}u + e{su}t + e{tu}s = 0 .
(IV.M )
(3.14)
s=1 t=s+1 u=t+1
Now we come to the critical argument of the proof. We claim that the following equation holds:
µX
¶
µ n−2
µ
¶¶
n
n
n
X X
X n−1
X X
e{st}u = 2
e{st}u + e{su}t + e{tu}s
.
s=1 t∈Adj(s)
u=1,
u6=s,t
(3.15)
s=1 t=s+1 u=t+1
Notice that the left-hand and right-hand sides of this equation consist only of coefficients e{st}u with s, t, u
distinct. Consider any such coefficient e{st}u . Notice that e{st}u appears exactly once on the right side of the
equation. Furthermore, either e{st}u appears exactly twice on the left side of this equation (since s ∈ Adj(t)
implies t ∈ Adj(s)), or e{st}u = 0 (since the edge {s, t} does not exist in the graph). Therefore, Eq. 3.15 is
proven. Applying this result (and Eq. 3.14) to Eq. 3.13 gives us the following:
X
e{st}s = 0 .
1≤s,t≤n
s6=t
But Eq. 3.16 contradicts Eq. 3.11 (1 = 0), thus there can be no certificate of degree less than four.
(3.16)
24
J.A. De Loera, J. Lee, S. Margulies, S. Onn
It is important to note that when we try to construct certificates of degree four or greater, the equations
for the degree-6 monomials become considerably more complicated. In this case, the edge polynomials do
contribute monomials of degree six, which causes the above argument to break.
We conclude this subsection with a result that allows us to bound the degree of a minimum-degree
Nullstellensatz certificate of a particular graph, if that graph can be “reduced” to another graph whose
minimum-degree Nullstellensatz certificate is known.
Lemma 3.14.
1. If H is a subgraph of G , and H has a minimum-degree non-3-colorability Nullstellensatz certificate of
degree k , then G also has a minimum-degree non-3-colorability Nullstellensatz certificate of degree k .
2. Suppose that a non-3-colorable graph G can be transformed to a non-3-colorable graph H via a sequence
of identifications of non-adjacent nodes of G . If a minimum-degree non-3-colorability Nullstellensatz
certificate for H has degree k , then a minimum-degree non-3-colorable Nullstellensatz certificate for G
has degree at least k .
Proof.
Proof of 1: Since H is a subgraph of G , then any Nullstellensatz certificate for non-3-colorability of H is
also a Nullstellensatz certificate for non-3-colorability of G .
Proof of 2: Suppose that G has a Nullstellensatz certificate for non-3-colorability of degree less than k .
P
P
The certificate has the form 1 =
ai vi + b{ij} e{ij} where vi = x3i − 1 , e{ij} = x2i + xi xj + x2j , and
both ai and b{ij} denote polynomials of degree less than k . Since the certificate is an identity, the identity
must hold for all values of the variables. In particular, it must hold for every variable substitution xi = xj
when the nodes are non-adjacent. In this case, the variable reassignment (pictorially represented in Figure 7)
yields a Nullstellensatz certificate of degree less than k for the transformed graph H . Note that the parallel
edges that may arise are irrelevant to our considerations (see such examples in Figure 7). But this is in
3
3
3
2
2
4
2
2
5 2
0
1
4
5
0
0
0
1
3
1
5
1
1
Figure 7. Converting G (the 5-odd-wheel) to H (the 3-odd-wheel) via node identifications.
contradiction with the assumed degree of a minimum-degree certificate for H . Therefore, any certificate for
G must have degree at least k .
3.2.1. Cliques, Odd Wheels, and Nullstellensatz Certificates
Theorem 3.15. For Kn with n ≥ 4 , a minimum-degree Nullstellensatz certificate for non-3-colorability
has degree exactly four.
Proof. It is easy to see that K4 is a subgraph of K5 , which is a subgraph of K6 , and so on. The decision
problem of whether K4 is 3-colorable can be encoded by the system of equations
x31 − 1 = 0 , x32 − 1 = 0 , x21 + x1 x2 + x22 = 0 , x21 + x1 x3 + x23 = 0 , x21 + x1 x4 + x24 = 0 ,
x33 − 1 = 0 , x34 − 1 = 0 , x22 + x2 x3 + x23 = 0 , x22 + x2 x4 + x24 = 0 , x23 + x3 x4 + x24 = 0 ,
Expressing Combinatorial Problems by Polynomial Equations
25
with one equation per vertex and one equation per edge. Using the linear-algebra heuristic described at the
beginning of Section 3, we find that K4 has a degree-4 Nullstellensatz certificate for non-3-colorablility:
µ
¶
5
2
4
2
2
4 4
1 = (−x31 − 1)(x31 − 1) +
x4 − x34 x2 − x34 x3 − x34 x1 + x24 x2 x1 + x24 x3 x1 (x24 + x2 x4 + x22 )
9
9
9
9
9
9
µ
µ
¶
¶
2
1 4
2 3
1 3
2 2
1
1
2
+
x4 + x4 x2 − x4 x1 − x4 x2 x1 (x22 + x3 x2 + x23 ) +
x44 + x34 x2 + x34 x1 + x24 x2 x1 (x24 + x3 x4 + x23 )
9
9
9
9
9
9
9
9
µ
¶
µ
¶
2 4
1 3
1 4
1
3
3
4
2
2
2
2
+ − x4 + x4 x1 − x4 x1 + x1 (x4 + x1 x4 + x1 ) + x4 x2 (x2 + x1 x2 + x1 ) + − x4 − x34 x2 (x23 + x1 x3 + x21 ).
3
3
3
3
Because K4 has a degree-4 Nullstellensatz certificate as shown above, Kn for n ≥ 4 also has a degree-4
Nullstellensatz certificate via Lemma 3.14 (1).
The odd-wheels consist of an odd-cycle rim, with a center vertex connected to all other vertices. It is
rather easy to see that no odd-wheel is 3-colorable. It is natural to ask about the degree of a minimumdegree Nullstellensatz certificate for non-3-colorablility.
Theorem 3.16. Given any odd-wheel, the degree of a minimum-degree Nullstellensatz certificate for non3-colorability is four.
Proof. First, we will prove that there exists a certificate of degree four for the n-th odd-wheel. We begin
by displaying a degree-4 certificate for the 3-odd-wheel:
µ
¶
4 4 5 3
2
4
2
2
1=
x1 − x1 x2 − x31 x3 − x31 x0 + x21 x2 x0 + x21 x3 x0 (x21 + x2 x1 + x22 )+
9
9
9
9
9
9
¶
µ
1 3
2 2
1
1 4 2 3
x1 + x1 x2 − x1 x0 − x1 x2 x0 (x22 + x3 x2 + x23 ) + x31 x2 (x22 + x0 x2 + x20 )+
9
9
9
9
3
µ
¶
2 4 1 3
1 3
2 2
1
x1 + x1 x2 + x1 x0 + x1 x2 x0 (x21 + x3 x1 + x23 ) + x41 (x21 + x0 x1 + x20 )+
9
9
9
9
3
µ
¶
1 4 1 3
(3.17)
− x1 − x1 x2 (x23 + x0 x3 + x20 ) + (−x31 − 1)(x31 − 1).
3
3
For now, we denote the non-3-colorability certificate for the 3-odd-wheel as follows:
1 = α{12} e{12} + α{23} e{23} + α{20} e{20} + α{13} e{13} + α{10} e{10} + α{30} e{30} + α1 v1 ,
where v1 = x31 − 1 , and e{ij} = x2i + xi xj + x2j and α1 and α{ij} denote polynomials of degree four in
R[x0 , x1 , x2 , x3 ] . In Figure 8, we can see that the topological difference between the 3-odd-wheel and the
2
2
0
0
3
4
2
0
1
3 1
3
5
1
5
4
Figure 8. Here we show the evolution of the 3-odd-wheel to the 5-odd-wheel. This can be extended from the n-th odd wheel to
the (n + 2)-th odd wheel.
5-odd-wheel is that the edge (1, 3) is lost, and the vertices 4, 5 and associated edges (3, 4), (4, 5), (5, 1), (0, 4)
and (0, 5) are gained. We can exhibit an algebraic relation (or syzygy) as follows:
α{13} e{13} = α{13} e{15} + β{34} e{34} + β{45} e{45} + β{01} e{01} + β{03} e{03} + β{04} e{04}
26
J.A. De Loera, J. Lee, S. Margulies, S. Onn
+ β{05} e{05} ,
(3.18)
where β{ij} ∈ R[x0 , x1 , x2 , x3 , x4 , x5 ] and deg(β{ij} ) ≤ 4 . Note that the coefficients for e{13} and e{15} are
the same. Recall that
α{13} =
2 4 1 3
1
2
x1 + x1 x2 + x31 x0 + x21 x2 x0 .
9
9
9
9
Since α{13} does not contain the variable x3 , α{13} ∈ R[x0 , x1 , x2 ] and not in R[x0 , x1 , x2 , x3 ] . Therefore,
because there exists a degree-4 certificate for the 3-odd-wheel, we can simply use the above syzygy (Eq.
3.18) to substitute for the α{13} e{13} term in the degree-4 3-odd-wheel certificate (Eq. 3.17). The resulting
polynomial is a degree-4 certificate for the 5-odd-wheel where the coefficient for e{15} in the 5-odd-wheel
certificate is exactly the same as the coefficient for e{13} in the 3-odd-wheel certificate (both coefficients are
α{13} ). Thus, we can again use the syzygy of Eq. 3.18 (with the variable substitutions of x4 → x6 , x5 → x7 ,
and x3 → x5 ), and substitute for the α{13} e{15} term in the 5-odd-wheel certificate to obtain a degree-4
7-odd-wheel certificate. Thus, by induction, we obtain degree-4 certificates for all odd-wheels. It remains for
us to show that such a syzygy exists.
The special syzygy was found via the linear algebra heuristic described at the beginning of Section 3 and is
listed below. Note that the coefficients for e{13} and e{15} are indeed identical and are equal to the coefficient
for e{13} in the 3-odd-wheel certificate we presented earlier (Eq. 3.17).
µ
0=−
¶
µ
¶
2 4
1
1
2
1
1
2
2 4
x1 + x31 x2 + x31 x0 + x21 x2 x0 (x21 + x3 x1 + x23 ) +
x1 + x31 x2 + x31 x0 + x21 x2 x0 (x21 + x5 x1 + x25 )
9
9
9
9
9
9
9
9
|
{z
}
|
{z
}
|
{z
}
|
{z
}
e{13}
e{15}
µ
+
µ
+
α{13}
¶
(x23 + x3 x4 + x24 )
|
{z
}
e{34}
2 4
2
1
1
1
1
1
1
2
x − x21 x2 x0 − x21 x2 x4 + x21 x0 x4 − x1 x2 x3 x0 + x1 x2 x3 x4 − x1 x2 x20 + x1 x2 x24 − x40
9 1
9
9
9
9
9
9
9
9
¶
1 3
1 4
1 3
1
3
2
2
+ x0 x4 − x4 + x4 x5 − x4 x5 (x4 + x4 x5 + x5 )
9
9
9
9
|
{z
}
−
µ
+
e{45}
1
2
5
1
2
2
1
2
2
x1 x3 x20 − x3 x0 x24 − x1 x23 x0 − x21 x3 x0 + x21 x4 x5 + x20 x4 x5 − x1 x4 x25 + x23 x0 x4 + x2 x3 x24
3
9
9
3
9
9
9
9
9
1 2
1 2
2 3
2 3
1 2
2 2 2
2 2 2
4
2
4
2
+ x1 x2 x3 − x1 x2 x5 + x1 x3 − x1 x5 + x1 x0 x5 − x1 x0 + x1 x4 − x1 x3 x4 − x1 x3 x0 x4 − x1 x0 x4 x5
9
9
9
9
9
9
9
9
3
9
5
4
1
1
2
2
1
1
2
1
2
2
2
2
3
2
2
2
− x1 x0 x4 − x1 x0 x4 − x1 x0 x5 − x1 x4 x5 − x1 x0 + x2 x3 x0 + x2 x3 x4 − x2 x3 x5 + x2 x0 x24 + x2 x3 x0 x4
9
9
9
9
9
9
9
9
9
3
¶
1
1
4 3
1 4
1 3
2 2 2
2 2 2
1
3
3
2
2
− x2 x3 x0 x5 + x2 x4 − x3 x0 − x3 − x3 x4 + x3 x4 + x0 x5 − x0 x4 (x0 + x0 x1 + x1 )
9
9
9
3
9
9
9
9
|
{z
}
−
µ
+
µ
+
e{01}
2 4
1
4
4
1
1
1
1
5
5
2
x + x31 x2 + x31 x0 + x31 x4 − x21 x2 x4 + x21 x23 + x21 x3 x0 + x21 x3 x4 + x21 x20 + x21 x0 x4 + x21 x24
9 1
9
9
9
9
3
9
9
9
9
9
2
1
1
1
1
2
1
1 2 2
2
2
2
− x1 x2 x0 − x1 x2 x0 x4 − x1 x2 x0 x5 + x1 x2 x4 x5 + x1 x3 x0 + x1 x3 x0 + x1 x3 x0 x4 + x3 x0
9
9
9
9
3
9
3
3
¶
1
1
2
2 4
2 3
3
2
2
2
2
− x3 x0 − x3 x0 x4 − x3 x0 x4 − x0 − x0 x4 (x0 + x0 x3 + x3 )
9
9
9
9
9
|
{z
}
e{03}
1 3
2
1
4
1
1
1
1
2
x x5 − x21 x2 x3 + x21 x2 x5 − x21 x23 − x1 x2 x3 x4 + x1 x2 x20 − x1 x2 x24 + x1 x3 x20 + x1 x3 x0 x4
9 1
9
9
9
9
9
9
9
9
1
1
1
1
1
2
1
1
1
+ x1 x30 + x1 x20 x4 + x1 x20 x5 + x2 x3 x0 x5 + x2 x3 x25 + x33 x0 + x23 x0 x4 − x23 x24 + x3 x30
3
9
9
9
9
9
9
9
3
¶
1
1
2 4
2
3
2
2
+ x3 x0 x4 − x3 x4 + x0 (x0 + x0 x4 + x4 )
9
9
9
{z
}
|
e{04}
µ
+
α{13}
1
1
1
2
2
1
1
2 3
x x0 + x1 x2 x0 x5 − x1 x2 x4 x5 − x1 x3 x20 − x1 x3 x0 x4 − x2 x30 − x2 x20 x4 + x44
9 1
9
9
9
9
9
9
9
−
1 3
1
1
1
1
2
1
1
1
x x2 + x31 x4 + x21 x2 x3 + x21 x2 x4 − x21 x20 + x1 x2 x3 x0 − x1 x2 x3 x4 + x1 x2 x20 − x1 x2 x24
9 1
9
9
9
9
9
9
9
9
Expressing Combinatorial Problems by Polynomial Equations
−
1
1
1
1
1
1
1
1
x1 x30 + x1 x20 x4 − x2 x3 x0 x4 − x2 x3 x24 − x0 x24 x5 − x0 x4 x25 + x24 x25 + x4 x35
9
9
9
9
9
9
9
9
27
¶
(x20 + x0 x5 + x25 ) .
|
{z
}
e{05}
Finally, the reader may easily observe that Theorem 1.2, Part 2 follows directly from Theorem 3.15,
Theorem 3.16, and Lemma 3.14.
3.3. Nullstellensatz Certificates for Other Non-3-Colorable Graphs
With the aid of a computer, we searched many families of non-3-colorable graphs, hoping to find explicit
examples with growth in the certificate degree. Every graph we have investigated so far has a Nullstellensatz
certificate of degree four. This suggests that examples with degree growth are rare for graphs with few
vertices, and that many graphs have short proofs of non-3-colorability. In Figure 9, we describe the Jin and
Grötzch graphs, and in Figure 10, we describe the “Flower” family. Kneser graphs are described in most
graph theory books. In Table 1, we present a sampling of the many graphs we tried during our computational
experiments. Note that we often used our probabilistic linear algebra algorithm, selecting p = .4 as a likely
threshold for feasibility.
Graph
vertices
edges
row
col
p
deg
flower 8
16
32
51819
49516
.4
4
flower 10
20
40
178571
362705
1
4
flower 11
22
44
278737
278844
.5
4
flower 13
26
52
629666
495051
.4
4
flower 14
28
56
923580
705536
.4
4
flower 16
32
64
1979584
1674379
.4
4
flower 17
34
68
2719979
2246535
.4
4
flower 19
38
76
4862753
3850300
.5
4
kneser-(6,2)
15
45
39059
68811
.5
4
kneser-(7,2)
21
105
230861
558484
.5
4
kneser-(8,2)
28
210
1107881
3307971
.5
4
kneser-(9,2)
36
378
1107955
3304966
.5
4
kneser-(10,2)
45
630
15,567,791
36,785,283
.5
4
jin graph
12
24
12168
13150
.4
4
Grötzsch
11
20
7903
8109
.4
4
G + {(3, 4)}
12
24
12,257
13,091
.4
4
G + {(7, 12)}
12
24
12,201
13,085
.4
4
G + {(1, 8)}
12
24
12,180
13,124
.4
4
G + {(3, 4), (12, 7)}
12
25
12,286
13,804
.4
4
Table 1. Experimental investigations for Flowers, Kneser, the Jin graph and the Grötzch graph. Here G denotes the
uniquely-colorable graph displayed in Figure 9.
A uniquely 3-colorable graph is a graph that can be colored with three colors in only one way, up to
permutation of the color labels. Figure 9 displays a uniquely 3-colorable triangle-free graph [6]. Since the
graph is uniquely 3-colorable, the addition of a single edge between two similarly-colored vertices will result
28
J.A. De Loera, J. Lee, S. Margulies, S. Onn
1
1
2
2
8
5
6
9
7
12
10
7
12
3
11
8
11
10
9
4
4
6
3
5
Figure 9. These graphs (from left to right) are (1) a uniquely 3-colorable graph, labeled with its unique 3-coloring [6], (2) the
Grötzsch graph, and (3) the Jin graph.
in a new non-3-colorable graph. Table 1 also details these experiments. Finally, we investigated all non-3colorable graphs on six vertices or less: every one has a Nullstellensatz certificate of degree four.
Figure 10. 3, 4 and 5 flowers (left to right). Note that the 3-flower is 3-colorable, whereas the 4 and 5 flowers are
non-3-colorable. It is easy to see that only flowers that are multiples of 3 are 3-colorable.
References
[1] N. Alon, Combinatorial Nullstellensatz, Combin. Prob. and Comput., 8, (1999), 7–29.
[2] N. Alon and M. Tarsi, Colorings and orientations of graphs, Combinatorica, 12, (1992), 125–134.
[3] E. Babson, S. Onn and R.R. Thomas, The Hilbert zonotope and a polynomial time algorithm for universal
Gröbner bases, Advances in Applied Mathematics, 30, (2003), 529–544.
[4] D.A. Bayer, “The Division Algorithm and the Hilbert Scheme,” Ph.D. Thesis, Harvard University, (1982).
[5] S. Buss and T. Pitassi, Good degree bounds on nullstellensatz refutations of the induction principle, IEEE
Conference on Computational Complexity, (1996), 233–242.
[6] C.-Y. Chao and Z. Chen, On uniquely 3-colorable graphs, Discrete Mathematics, 112, (1993), 374–383.
[7] D. Cox, J. Little and D. O’Shea, “Ideals,Varieties and Algorithms,” Springer Undergraduate Texts in Mathematics, Springer-Verlag, New York, (1992).
[8] D. Cox, J. Little and D. O’Shea, “Using Algebraic Geometry,” Springer Verlag Graduate Texts in Mathematics,
vol. 185, (1998).
[9] J.A. De Loera, Gröbner bases and graph colorings, Beitrage zur Algebra und Geometrie, 36(1), (1995), 89–96.
[10] S. Eliahou, An algebraic criterion for a graph to be four-colourable, Aportaciones Matemáticas, Soc. Matemática
Mexicana, Notas de Investigacion, 6, (1992), 3–27.
[11] K.G. Fischer, Symmetric polynomials and Hall’s theorem, Discrete Math, 69(3), (1988), 225–234.
[12] M. Garey and D. Johnson, “Computers and Intractability: A Guide to the Theory of NP-Completeness,” W.H.
Freeman and Company, Copyright 1979 Bell Telephone Laboratories, Incorporated.
[13] J. Griggs, C. Grinstead and D. Guichard, The number of maximal independent sets in a connected graph ,
Discrete Mathematics, 68(2-3), (1988), 211–220.
[14] J. Harant and I. Schiermeyer, On the independence number of a graph in terms of order and size, Discrete
Mathematics, 232(1-3), (2001), 131–138.
[15] C.J. Hillar and T. Windfeldt, “An algebraic characterization of uniquely vertex colorable graphs,” manuscript
2006, see http://www.math.tamu.edu/~chillar/files/uniquegraphcolorings.pdf
[16] P. Impagliazzo, P. Pudlák and J. Sgall, Lower bounds for polynomial calculus and the Groebner basis algorithm,
Computational Complexity, 8, (1999), 127–144.
[17] G. Jin, Triangle-free four-chromatic graphs, Discrete Mathematics, 145, (1995), 151–170.
[18] J. Kollar, Sharp effective Nullstellensatz, J. of the AMS, 1(4), (1988), 963–975.
[19] M. Kreuzer and L. Robbiano, “Computational Commutative Algebra I,” Springer Verlag, Heidelberg, (2000).
Expressing Combinatorial Problems by Polynomial Equations
29
[20] J.B. Lasserre, Polynomials nonnegative on a grid and discrete optimization, Transactions of the AMS, 354(2),
(2001), 631–649.
[21] M. Laurent, Semidefinite representations for finite varieties, Mathematical Programming, 109, (2007), 1–26.
[22] M. Laurent and F. Rendl, Semidefinite programming and integer programming. In: “Handbook on Discrete
Optimization,” K. Aardal, G. Nemhauser, R. Weismantel (eds.), Elsevier B.V., (2005), 393–514.
[23] S.R. Li and W.W. Li, Independence number of graphs and generators of ideals, Combinatorica, 1, (1981), 55–61.
[24] L. Lovász, Stable sets and polynomials, Discrete Mathematics, 124, (1994), 137–153.
[25] L. Lovász, Oberwolfach meeting on Geometric Convex Combinatorics, Mathematisches Forschungsinstitut Oberwolfach, Germany, June 2002.
[26] S. Margulies, “Combinatorics, Computer Algebra and Complexity: Gröbner Bases and NP-Complete Problems,”
UC Davis Ph.D. dissertation, in preparation, (2008).
[27] Y. Matiyasevich, A criteria for colorability of vertices stated in terms of edge orientations, (in Russian), Discrete
Analysis (Novosibirsk), 26, (1974), 65–71.
[28] Y. Matiyasevich, Some algebraic methods for calculation of the number of colorings of a graph, (in Russian),
Zapiski Nauchnykh Seminarov POMI, 293, (2001), 193–205 (available via www.pdmi.ras.ru).
[29] M. Mnuk, Representing graph properties by polynomial ideals, In V. G. Ganzha, E. W. Mayr, and E. V.
Vorozhtsov, editors, “Computer Algebra in Scientific Computing, CASC 2001.” Proceedings of the Fourth International Workshop on Computer Algebra in Scientific Computing, Konstanz, Springer-Verlag, (2001), 431–444.
[30] S. Onn, Nowhere-zero flow polynomials, Journal of Combinatorial Theory, Series A, 108, (2004), 205–215.
[31] P. Parrilo, Semidefinite programming relaxations for semialgebraic problems,” Mathematical Programming Ser.
B, 96(2), (2003), 293–320.
[32] P. Parrilo, “An explicit construction of distinguished representations of polynomials nonnegative over finite sets,”
IfA Technical Report AUT02-02, (2002).
[33] W. Schnyder, Planar graphs and poset dimension, Order, 5, (1989), 323–343.
[34] A. Schrijver, “Theory of Linear and Integer Programming,” Wiley-Interscience Series in Discrete Mathematics
and Optimization, John Wiley & Sons, Chichester, West Sussex, England, 1986.
[35] S.E. Shauger, Results on the Erdös-Gyárfás conjecture in K1,m -free graphs, “Proceedings of the Twenty-ninth
Southeastern International Conference on Combinatorics Graph Theory and Computing (Boca Raton, FL,
1998).” Congr. Numer., 134, (1998), 61–65.
[36] A. Simis, W. Vasconcelos and R. Villarreal, On the ideal theory of graphs, J. Algebra, 167(2), (1994), 389–416.
[37] M. Yannakakis, Expressing combinatorial optimization problems by linear programs, J. Comput. Syst. Sci., 43(3),
(1991), 441–466.